BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes withd= δ, wheredis the minimum distance of aq-ary BCH code of lengthn, designed distance δ, and offsetb, denoted by C(q,n,δ,b). For primitive BCH codes, we obtain infinite families of BCH codes over F3and F4satisfyingd= δ, where δ 2 {5, 6, 7, 8}. Moreover, we construct several infinite families ofq-ary BCH codes withd= δ, where 2 ≤ δ ≤q−1. For δ =qt+1, we prove that the BCH code C(q,qm−1,qt+1,1)hasd= δ for allmsatisfyingm≡ 0 (modpt), wherepdenotes the characteristic of Fq. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of C(q,qm−1,qt+1,1)is always equal to its Bose distancedB. Our result confirms this conjecture for the casem≡ 0 (modpt). For non-primitive BCH codes, we construct a family of BCH codes C(q, qp−1/λ ,p+1,1)withd= δ =p+ 1, wherepis an odd prime,q = pewithp∤eand λ |q− 1.

Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary andp-ary cyclic codes for odd primepin the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over Fqfor each evenq≥ 4. In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; seven infinite families of repeated-root cyclic codes with minimum distance 6 or 8 or 10 are presented; and two infinite families of repeated-root binary cyclic codes with parameters [2n,k,d≥ (n− 1)/ log2n], wheren= 2m− 1 andk≥n, are constructed. In addition, 26 repeated-root cyclic codes of length up to 254 over Fqforqϵ {2, 4, 8} with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.

In this paper, we present an antiGriesmer lower bound on the subcode support weights of projective linear codes. This bound is a lower bound on the maximumr-dimensional subcode support weights of projective linear codes. Based on the r-dimensional subcode support weight distributions (r-SSWDs) of linear codes, we compute ther-SSWDs for their simplex complementary codes. Then we construct several infinite families of distance-optimal codes meeting the ℓ-generalized Hamming weight Griesmer bound. These codes do not achieve the Griesmer bound for thej-generalized Hamming weight, where 1 ≤j< ℓ. Moreover, subcode support weight distributions of these optimal linear codes are determined.

Binary self-dual cyclic codes have been studied since the classical work of Sloane and Thompson published in IEEE Trans. Inf. Theory, vol. 29, 1983. Twenty five years later, an infinite family of binary self-dual cyclic codes with lengths$n_{i}$and minimum distances$d_{i} \geq \frac {1}{2} \sqrt {n_{i}+2}$was presented in a paper of IEEE Trans. Inf. Theory, vol. 55, 2009. However, no infinite family of Euclidean self-dual binary cyclic codes whose minimum distances have the square-root lower bound and no infinite family of Euclidean self-dual nonbinary cyclic codes whose minimum distances have a lower bound better than the square-root lower bound are known in the literature. In this paper, an infinite family of Euclidean self-dual cyclic codes over the fields${\mathrm { F}}_{2^{s}}$with a square-root-like lower bound is constructed. An infinite subfamily of this family consists of self-dual binary cyclic codes with the square-root lower bound. Another infinite subfamily of this family consists of self-dual cyclic codes over the fields${\mathrm { F}}_{2^{s}}$with a lower bound better than the square-root bound for$s \geq 2$. Consequently, two breakthroughs in coding theory are made in this paper. An infinite family of self-dual binary cyclic codes with a square-root-like lower bound is also presented in this paper. An infinite family of Hermitian self-dual cyclic codes over the fields${\mathrm { F}}_{2^{2s}}$with a square-root-like lower bound and an infinite family of Euclidean self-dual linear codes over${\mathrm { F}}_{q}$with$q \equiv 1 \pmod {4}$with a square-root-like lower bound are also constructed in this paper.

For a given linear code$\mathcal {C}$of length n over${\mathrm {GF}}(q)$and a nonzero vector u in${\mathrm {GF}}(q)^{n}$, Sun, Ding and Chen defined an extended linear code$\overline {\mathcal {C}}({\mathbf {u}})$of$\mathcal {C}$, which is a generalisation of the classical extended code$\overline {\mathcal {C}}(-{\mathbf {1}})$of$\mathcal {C}$and called the second kind of an extended code of$\mathcal {C}$(see Finite Fields Appl., vol. 96, 102401, 2024 and Discrete Math., vol. 347, no. 9, 114080, 2024). They developed some general theory of the extended codes$\overline {\mathcal {C}}({\mathbf {u}})$and studied the extended codes$\overline {\mathcal {C}}({\mathbf {u}})$of several families of linear codes, including cyclic codes, projective two-weight codes, nonbinary Hamming codes, and a family of reversible MDS cyclic codes. The objective of this paper is to investigate the extended codes$\overline {\mathcal {C}}({\mathbf {u}})$of MDS codes$\mathcal {C}$over finite fields. The main result of this paper is that the extended code$\overline {\mathcal {C}}({\mathbf {u}})$of an MDS$[n,k]$code$\mathcal {C}$remains MDS if and only if the covering radius$\rho (\mathcal {C}^{\bot })=k$and the vector u is a deep hole of the dual code${\mathcal {C}}^{\perp } $. As applications of this main result, an equivalent statement of MDS Conjecture is presented, the extended codes of the GRS codes and extended GRS codes are investigated, and the covering radii and some deep holes of several families of MDS codes are also determined.

Constacyclic codes over finite fields are an important class of linear codes as they contain distance-optimal codes and linear codes with best known parameters. They are interesting in theory and practice, as they have the constacyclic structure. In this paper, an infinite class of q-ary negacyclic codes of length$(q^{m}-1)/2$and an infinite class of q-ary constacyclic codes of length$(q^{m}-1)/(q-1)$are constructed and analyzed. As a by-product, two infinite classes of ternary negacyclic self-dual codes with a square-root-like lower bound on their minimum distances are presented.

Constacyclic codes contain cyclic codes as a subclass and have nice algebraic structures. Constacyclic codes have theoretical importance, as they are connected to a number of areas of mathematics and outperform cyclic codes in several aspects. Negacyclic codes are a subclass of constacyclic codes and are distance-optimal in many cases. However, compared with the extensive study of cyclic codes, negacyclic codes are much less studied. In this paper, several families of ternary negacyclic codes and their duals are constructed and analysed. These families of negacyclic codes and their duals contain distance-optimal codes and have very good parameters in general. The duals of three families of ternary negacyclic codes presented in this paper are distance-optimal.

A linear code with parameters [n,k,n−k+1] is called a maximum distance separable (MDS for short) code. A linear code with parameters [n,k,n−k] is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a [q+1,2u−1,q−2u+3] MDS code for eachuwith 1 ≤u≤ ⌊q+1/2⌋, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.

Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is motivated by the punctured Dilix cyclic codes and the second class is motivated by the punctured generalised Reed-Muller codes. The two classes of constacyclic codes contain optimal linear codes. The parameters of the two classes of constacyclic codes are analysed and some open problems are presented in this paper.

Binary cyclic codes with parameters$[n,(n+1)/2, d\geq \sqrt {n}]$are very interesting, as their minimum distances have a square-root bound. The binary quadratic residue codes and the punctured binary Reed-Muller codes of order$(m-1)/2$for odd$m$are two infinite families of binary cyclic codes with such parameters. The objective of this paper is to present and analyse an infinite family of binary BCH codes${\mathcal {C}}(m)$with parameters$[2^{m}-1,2^{m-1},d]$whose minimum distance$d$much exceeds the square-root bound when$m \geq 11$is a prime. The binary BCH code${\mathcal {C}}(3)$is the binary Hamming code and distance-optimal. The binary BCH code${\mathcal {C}}(5)$has parameters$[{31,16,7}]$and is distance-almost-optimal. The binary BCH code${\mathcal {C}}(7)$has parameters$[{127,64,21}]$and has the best known parameters. In addition, there is no known$[2^{m}-1,2^{m-1}]$binary cyclic code whose minimum distance is better than the minimum distance of this binary BCH code${\mathcal {C}}(m)$with parameters$[2^{m}-1,2^{m-1}]$for any odd prime$m$.

Cyclic codes are important in theory, as they are closely related to a number of areas of mathematics. Cyclic codes are also important in practice, as they have efficient encoding and decoding algorithms. An infinite family of cyclic codes over GF(q) is said to have linearly-best-known parameters if for any [n,k,d] codeCin this family, there is no known [n,k,d′] linear code over GF(q) such thatd′ >d. An infinite family of cyclic codes over GF(q) is said to have cyclicly-best-known parameters if for any [n,k,d] codeCin this family, there is no known [n,k,d′] cyclic code over GF(q) such thatd′ >d. It is very rare to see an infinite family of binary cyclic codes with cyclicly-best-known parameters whose duals codes have also cyclicly-best-known parameters. The objective of this paper is to study such family of binary cyclic codes of length 2m– 1 and dimension 2m– 1 –m(m– 1)/2, denoted byC(2,m,2), and their dual codesC⊥(2,m,2). The weight distribution ofC⊥(2,m,2)is settled and the parameters ofC(2,m,2)are investigated in this paper. A larger family of binary cyclic codesC(2,m,r)and their duals are also constructed and studied in this paper, where 0 ≤r≤m– 1.

The construction of self-dual codes with large minimum distances has been an active topic in coding theory. The construction and classification of extremal self-dual codes over small fields are related to other fields of mathematics, such as lattices and invariant theory as well as combinatorialt-designs. It is well-known thatq-ary self-dual cyclic codes exist only whenqis an even prime power andq-ary self-dual negacyclic codes exist for any odd prime powerq. In 2009 a family of binary self-dual cyclic codes with lengthsniand minimum distancesdi≥ 1/2√ni, wherenigoes to the infinity ifigoes to the infinity, was constructed. In this paper, we construct several families ofq-ary self-dual negacyclic codes of lengthsnwith their minimum distances larger than or equal ton1/2for various lengthsnand any given odd prime powerq. Whenq∈ {3, 5} and the length is small, the minimum distances of the constructed self-dual negacyclic codes are comparable with these self-dual codes with largest known minimum distances in the literature.

As a special subclass of cyclic codes, BCH codes are usually among the best cyclic codes and have wide applications in communication and storage systems and consumer electronics. Let$\mathcal C$be a$q^{2}$-ary BCH code of length$n$with respect to an$n$-th primitive root of unity$\beta $over an extension field of$\Bbb F_{q^{2}}$, and let$\mathcal C^{\perp H}$denote its Hermitian dual code, where$q$is a prime power. If both$\mathcal {C}$and$\mathcal C^{\perp H}$are a BCH code with respect to an$n$-th primitive root of unity$\beta $, then$\mathcal C$is called aHermitian dually-BCH code. The objective of this paper is to derive a necessary and sufficient condition for ensuring that two classes of narrow-sense BCH codes are Hermitian dually-BCH codes. As by-products, lower bounds on the minimum distances of the Hermitian dual codes of these BCH codes are developed, which improve the lower bounds documented in IEEE Trans. Inf. Theory, vol. 68, no. 2, pp. 953-964, 2022, in some cases.

Interplay between coding theory and combinatorial$t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of 3-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of 3-designs has been reported in the literature. This is the main motivation of this paper. Let$q=p^{m}$, where$p$is an odd prime and$m \geq 2$is an integer. The objective of this paper is to present an infinite family of cyclic codes over${\mathrm {GF}}(q)$supporting an infinite family of 3-designs and two infinite families of negacyclic codes over${\mathrm {GF}}(q^{2})$supporting two infinite families of 3-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over${\mathrm {GF}}(q)$are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over${\mathrm {GF}}(4)$supporting a 4-design and seven open problems are also presented in this paper.

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length$n$over$\mathbb {F}_{q}$is always relative to an$n$-th primitive root of unity$\beta $in an extension field of$\mathbb {F}_{q}$, and is called a dually-BCH code if its dual is also a BCH code relative to the same$\beta $. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances$\delta $will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances$\delta $. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.

Recently, subfield codes of geometric codes over large finite fields${\mathrm {GF}}(q)$with dimension 3 and 4 were studied and distance-optimal subfield codes over${\mathrm {GF}}(p)$were obtained, where$q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of$[q+1, 2, q]$MDS codes over${\mathrm {GF}}(q)$, and then studies the subfield codes over${\mathrm {GF}}(p)$of some of the$[q+1, 2,q]$MDS codes over${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over${\mathrm {GF}}(p)$are obtained, and several families of nearly optimal codes over${\mathrm {GF}}(p)$are produced. Several open problems are also proposed in this paper.

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field${\mathrm {GF}}(q)$with length$q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some$q$-ary BCH codes with length$q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of${\mathrm {PGL}}(2, p^{m})$-invariant codes over${\mathrm {GF}}(p^{h})$is completed. As an application of this result, the$p$-ranks of all incidence structures invariant under the projective general linear group${\mathrm {PGL}}(2, p^{m})$are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

The binary quadratic-residue codes and the punctured Reed-Muller codes${\mathcal {R}}_{2}((m-1)/2, m))$are two families of binary cyclic codes with parameters$[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length$2^{m}-1$and dimension near$2^{m-1}$with good minimum distances. When$m \geq 3$is odd, the codes become a family of duadic codes with parameters$[2^{m}-1, 2^{m-1}, d]$, where$d \geq 2^{(m-1)/2}+1$if$m \equiv 3 \pmod {4}$and$d \geq 2^{(m-1)/2}+3$if$m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes.

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of$[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$MDS codes for$1 \leq u \leq 2^{m-1}$, which are cyclic, reversible and BCH codes over${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even$m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented.

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some$p$-ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and$p$-ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes.

Let m be an even positive integer. A Boolean bent function f on F(2m-1)×F2is called a cyclic bent function if for any a≠b∈F(2m-1) and ε∈F2, f( ax1,x2)+f( bx1,x2+ε) is always bent, where x1∈F(2m-1),x2∈F2. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on F(2m-1)×F2and their applications. The first objective of this paper is to establish a link between quadratic cyclic bent functions and a special type of prequasifields, and construct a class of quadratic cyclic bent functions from the Kantor-Williams prequasifields. The second objective is to use cyclic bent functions to construct families of optimal sequences. The results of this paper show that cyclic bent functions have nice applications in several fields such as coding theory, symmetric cryptography, and CDMA communication.

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension m and length less than or equal to m(m+1)/2 are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes.

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper.

The question as to whether there exists an infinite family of near MDS codes holding an infinite family of t-designs for t ≥ 2 was answered in the recent paper [Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory 66(9) (2020)], where an infinite family of near MDS codes holding an infinite family of 3-designs and an infinite family of near MDS codes holding an infinite family of 2-designs were presented, but no infinite family of linear codes holding an infinite family of 4-designs was presented. Hence, the question as to whether there is an infinite family of linear codes holding an infinite family of 4-designs remains open for 71 years. This paper settles this longstanding problem by presenting an infinite family of BCH codes of length 2m+1+1 over GF(22m+1) holding an infinite family of 4-(22m+1+ 1, 6, 22m- 4) designs. This paper also provides another solution to the first question, as some of the BCH codes presented in this paper are also near MDS. Moreover, an infinite family of linear codes holding the spherical geometry design S(3, 5, 4m+ 1) is presented. The new direction of searching for t-designs with elementary symmetric polynomials will be further advanced.

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f)={ Tr(af(x)+bx)x ∈ \mathbb Fqm*: a,b ∈ \mathbb Fqm }, where q is a prime power, \mathbb Fqm* = \mathbb Fqm \{0}, Tr is the trace function from \mathbb Fqm to \mathbb Fq, and f(x) is a function from \mathbb Fqm to \mathbb Fqm with f(0)=0. Almost bent functions, quadratic functions and some monomials on \mathbb F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.

An [n, k, n - k + 1] linear code is called an MDS code. An [n, k, n - k] linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system S(5, 6, 12) with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for t ≥ 2 remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over GF(3s) holding an infinite family of 3-designs and an infinite family of near MDS codes over GF(22s) holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal.

The binary Hamming codes with parameters [2m-1, 2m-1- m, 3] are perfect. Their extended codes have parameters [2m, 2m- 1 - m, 4] and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters [2m+s+ 2s- 2m, 2m+s+ 2s- 2m- 2m -2, 4], which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters [2m+2, 2m-2m, 6]. Both classes of binary linear codes have new parameters.

In this paper, a family of [q2- 1, 4, q2- q - 2] cyclic codes over Fqmeeting the Griesmer bound is presented. Their duals are [q2- 1, q2- 5,4] almost MDS codes and are optimal with respect to the sphere-packing bound. The q0-ary subfield codes of this family of cyclic codes are also investigated, where q0is any prime power such that q is power of q0. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of [q2, 4, q2- q - 1] linear codes over Fqmeeting the Griesmer bound is presented. Their duals are [q2, q2- 4, 4] almost MDS codes and are optimal with respect to the sphere-packing bound. The q0-ary subfield codes of this family of linear codes are also investigated, where q0is any prime power such that q is power of q0. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper.

Boolean functions, coding theory and t-designs have close connections and interesting interplay. A standard approach to constructing t-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting t-designs. However, some linear codes hold t-designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a t-transitive or t-homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new t-designs. To this end, a general theory for punctured and shortened codes of linear codes supporting t-designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially δ-uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system S(2, 4, 2n) are produced in this paper.

Ovoids in PG(3, GF(q)) have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families of ovoids are known. The first is the elliptic quadrics and the second is the Tits ovoids. It is known that an ovoid in PG(3, GF(q)) corresponds to a [q2+ 1, 4, q2- q] code over GF(q), which is called an ovoid code. The objectives of this paper are to develop the general theories of subfield codes and to study the subfield codes of the two families of ovoid codes. The dimensions, minimum weights, and the weight distributions of the subfield codes of the elliptic quadric codes and Tits ovoid codes are settled. The parameters of the duals of these subfield codes are also studied. Some of the codes presented in this paper are optimal, and some are distance-optimal. The parameters of the subfield codes are new.

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (GF(22m), +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials.

In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, are preferred in secret sharing. In this paper, a necessary and sufficient condition for a binary linear code to be minimal is derived. This condition enables us to obtain three infinite families of minimal binary linear codes with Wmin/Wmax≤ 1/2 from a generic construction, where Wminand Wmax, respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined.

The projective special linear group PSL2(n) is 2-transitive for all primes n and 3-homogeneous for n = 3 (mod 4) on the set (0, 1, ... , n - 1, ∞). It is known that the extended odd-like quadratic residue codes are invariant under PSL2(n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n = 1 (mod 4), an infinite family of 3-designs for primes n = 3 (mod 4). To construct more t-designs with t ∈ (2, 31, one would search for other extended cyclic codes over finite fields that are invariant under the action of PSL2(n). The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under PSL2(n).

In practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing, and codebooks with small inner-product correlation are required. It is extremely difficult to construct codebooks achieving the Levenshtein bound. In this paper, two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented. These constructions produce complex codebooks asymptotically achieving the Levenshtein bound and codebooks asymptotically achieving the Welch bound. The codebooks presented in this paper have new parameters.

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes-a class of linear codes-have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over GF(q) with length n = (qm-1)/(q -1). Little is known about this class of BCH codes when q > 2. The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length n = (3m- 1)/2 are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.

Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.

A classical method of constructing a linear code over GF(q) with a t-design is to use the incidence matrix of the t-design as a generator matrix over GF(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.

Because of their efficient encoding and decoding algorithms, cyclic codes-an interesting class of linear codes- are widely used in communication systems, storage devices, and consumer electronics. BCH codes form a special class of cyclic codes, and are usually among the best cyclic codes. A subclass of good BCH codes is the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The main objective of this paper is to study the dimension and minimum distances of a subclass of the narrow-sense primitive BCH codes with design distance δ = (q - ℓ0)qm-ℓ1-1-1 for certain pairs (ℓ0, ℓ1), where 0 ≤ ℓ0q - 2 and 0 ≤ ℓ1m - 1. The parameters of other related classes of BCH codes are also investigated, and some open problems are proposed in this paper.

In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes. These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to their applications in consumer electronics, communication and data storage systems.

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms. Bose-Ray-Chaudhuri-Hocquenghem (BCH) codes form a subclass of cyclic codes and are very important in both theory and practice as they have good error-correcting capability and are widely used in communication systems, storage devices, and consumer electronics. However, the dimension and minimum distance of BCH codes are not known in general. The objective of this paper is to determine the Bose and minimum distances of a class of narrow-sense primitive BCH codes.

In this paper, a generic construction of codebooks based on binary codes is introduced. With this generic construction, a few previous constructions of optimal codebooks are extended, and a new class of codebooks almost meeting the Levenshtein bound is presented. Exponentially many codebooks meeting or almost meeting the Levenshtein bound from binary codes are obtained in this paper. The codebooks constructed in this paper have alphabet size 4. As a byproduct, three bounds on the parameters of binary codes are derived.

In this paper, we investigate the permutation property of the Dickson polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$. Due to a known result, it suffices to consider permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$, where $p$ is a prime and $t$ is a positive integer. We identify all permutation polynomials of $E_n(x, a)$ over $\mathbb{Z}_{p^t}$ for (I) $p=2$ and (II) $p$ is odd and $a$ is a square over $\mathbb{Z}_p$. For odd $p$ and nonsquares $a$ in $\mathbb{Z}_p$, we determine a large class (if not all) of permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$. A conjecture is also presented in this paper. If this conjecture is true, then all Dickson permutation polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$ are determined.

Zero-difference balanced (ZDB) functions integrate a number of subjects in combinatorics and algebra, and have many applications in coding theory, cryptography, and communications engineering. In this paper, three new families of ZDB functions are presented. The first construction gives ZDB functions defined on the abelian groups (GF(q1)×,...,×GF(qk),+) with new and flexible parameters. The other two constructions are based on 2-cyclotomic cosets and yield ZDB functions on \BBZn with new parameters. The parameters of optimal constant composition codes, optimal, and perfect difference systems of sets obtained from these new families of ZDB functions are also summarized.

Let m ≥ 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C(1,e) over Fp, which have parity-check polynomial m1(x)me(x), are presented by examining general conditions on the parameters p, m, and e, where mi(x) is the minimal polynomial of π-i over Fp for a primitive element π of Fpm. Furthermore, for p ≡ 3 (mod 4) and a positive integer e satisfying (pk+ 1) · e ≡ 2 (mod pm - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Σx∈FpmωTr(ax+bxe)and S(a, b, c) = Σx∈FpmωTr(ax+bxe+cxs), where s = (pm- 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C(1,e,s)with parity-check polynomial m1(x)me(x)ms(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C(1,e,s)has optimal minimum distance.

In this paper, a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced. It includes some earlier constructions of codebooks meeting the Levenstein bound as special cases. With this construction, two new families of codebooks achieving the Levenstein bound are obtained. The codebooks constructed in this paper could have a very small alphabet size.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, some monomials and trinomials over finite fields are employed to construct a number of families of cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Many of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding cyclic codes from monomials and trinomials are also presented.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms, compared with linear block codes. In this paper, seven classes of three-weight cyclic codes over \gf(p) whose duals have two zeros are presented, where p is an odd prime. The weight distributions of the seven classes of cyclic codes are settled. Some of the cyclic codes are optimal in the sense that they meet certain bounds on linear codes. The application of these cyclic codes in secret sharing is also considered.

As a subclass of linear codes, cyclic codes have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, five families of three-weight ternary cyclic codes whose duals have two zeros are presented. The weight distributions of the five families of cyclic codes are settled. The duals of two families of the cyclic codes are optimal.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters [3m-1, 3m-1-2m, 4] by Carlet, Ding, and Yuan in 2005. In this paper, almost perfect nonlinear monomials, and a number of other monomials over GF(3m) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of p-ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p2m-1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined.

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length n1n2and dimension (n1n2+ 1)/2 are presented and analyzed, where n1and n2are two distinct primes. Bounds on their minimum odd-like weight are also proved. Some of the codes presented in this paper are among the best cyclic codes.

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, the two-prime sequence is employed to construct several classes of cyclic codes over GF(q). Lower bounds on the minimum weight of these cyclic codes are developed. Some of the codes obtained are optimal or almost optimal. Thep-ranks of the twin-prime difference sets and a class of almost difference sets are computed.

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few more cases.

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few cases. The weight distributions of punctured versions of these codes are also determined for several special cases.

Binary sequences with low correlation have applications in communication systems and cryptography. Though binary sequences with optimal autocorrelation were constructed in the literature, no pair of binary sequences with optimal autocorrelation are known to have also best possible cross correlation. In this paper, new bounds on the cross correlation of binary sequences with optimal autocorrelation are derived, and pairs of binary sequences having optimal autocorrelation and meeting some of these bounds are presented. These new bounds are better than the Sarwate bounds on the cross correlation of binary sequences with optimal autocorrelation.

In communication systems, frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are used in FH-CDMA systems. In this paper, an earlier idea of constructing optimal sets of frequency hopping sequences is further investigated. New optimal parameters of sets of frequency hopping sequences are obtained with subcodes of the Reed-Solomon codes. Optimal sets of frequency hopping sequences are constructed with a class of irreducible cyclic codes. As a byproduct, the weight distribution of a subclass of irreducible cyclic codes is determined.

Sequences with optimal autocorrelation property are needed in certain communication systems and cryptography. In this paper, a construction of balanced quaternary sequences with periodN≡ 2 (mod 4) and optimal autocorrelation value and a construction of almost balanced binary sequences with periodN≡ 0 (mod 4) and optimal autocorrelation value are presented. Both constructions are a generalization of earlier ones.

Irreducible cyclic codes have been an interesting subject of study for a long time. Their weight distribution is known in only a few cases. In this paper, the weight distribution of the irreducible cyclic codes in a number of other cases is determined. The number of nonzero weights in the codes dealt with in this paper varies between one and four.

Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies in communication systems. Frequency hopping sequences are needed in frequency hopping code-division multiple-access (FH-CDMA) systems. In this paper, four algebraic and a combinatorial constructions of optimal sets of frequency hopping sequences with new parameters are presented, and a number of bounds on sets of frequency hopping sequences are described.

Constant composition codes are a special class of constant weight codes, and include permutation codes as a subclass. They have applications in communications engineering. In this correspondence, a generic construction of optimal constant composition codes using zero-difference balanced functions is introduced. It generalizes the earlier construction of optimal constant composition codes employing perfect nonlinear functions. In addition, two classes of optimal constant composition codes with new parameters are reported.

In communication systems, frequency-hopping spread spectrum and direct-sequence spread spectrum are two main spread coding technologies. Frequency-hopping sequences are needed in FH-CDMA systems. In this correspondence, three algebraic constructions of sets of optimal frequency-hopping sequences are presented. The parameters of these sets of frequency-hopping sequences are new and flexible.

Signal sets with the best correlation property are desirable in code-division multiple-access (CDMA) systems. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions and almost bent functions are employed to obtain (q2+q,q) signal sets. The signal sets derived from planar functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets constructed in this paper could have a very small alphabet size, and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor

Codebooks (also called signal sets) meeting the Welch bound on the maximum correlation amplitude are called MWBE codebooks and are desirable in code-division multiple-access systems. Two different but related constructions of MWBE codebooks from difference sets were developed by Xia and Ding recently. The objectives of this correspondence are to present a generic construction of MWBE codebooks that contains the previous two constructions as special cases and describe new MWBE codebooks that cannot be produced by the earlier two constructions.

In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several classes of authentication codes using certain classes of error-correcting codes are described. The authentication codes presented in this paper are better than known ones with comparable parameters. It is demonstrated that the construction is related to certain combinatorial designs, such as difference matrices and generalized Hadamard matrices

Frequency-hopping (FH) spread spectrum and direct-sequence spread spectrum are two main spread-coding technologies. Frequency-hopping sequences are needed in FH code-division multiple-access (CDMA) systems. In this correspondence, three classes of optimal frequency-hopping sequences are constructed with algebraic methods. The three classes are based on perfect nonlinear functions, power functions, and norm functions, respectively. Both individual optimal frequency-hopping sequences and optimal families of frequency-hopping sequences are presented.

In this correspondence, two classes of cyclotomic linear codes over GF(q) of order 3 are constructed and their weight distributions are determined. The two classes are two-weight codes and contain optimal codes. They are not equivalent to irreducible cyclic codes in general when q > 2.

We construct two families of authentication schemes using highly nonlinear functions on finite fields of characteristic 2. This leads to improvements on an earlier construction by Ding and Niederreiter if one chooses, for instance, an almost bent function as the highly nonlinear function

Codebooks (also called signal sets) meeting the Welch bounds are desirable in code-division multiple-access (CDMA) systems. In 2003, binary codebooks meeting Welch's bounds were constructed using difference sets by Ding, Golin, and Kloslashve. Recently, a generic construction of codebooks with cyclic difference sets meeting Welch's bound on the maximum cross-correlation amplitude was developed by Xia In this correspondence, the idea of Xia is extended, and related constructions of optimal codebooks with both cyclic and noncyclic difference sets are presented. These codebooks are optimal in the sense that they also meet this Welch bound. In addition, complex codebooks that almost meet Welch's bound on the maximum cross-correlation amplitude are also constructed with almost difference sets

In this correspondence, the weight distribution of a class of linear codes based on perfect nonlinear functions (also called planar functions) is determined. The class of linear codes under study are either optimal or among the best codes known, and have nice applications in cryptography.

Secret sharing has been a subject of study for over 20 years, and has had a number of real-world applications. There are several approaches to the construction of secret sharing schemes. One of them is based on coding theory. In principle, every linear code can be used to construct secret sharing schemes. But determining the access structure is very hard as this requires the complete characterization of the minimal codewords of the underlying linear code, which is a difficult problem in general. In this paper, a sufficient condition for all nonzero codewords of a linear code to be minimal is derived from exponential sums. Some linear codes whose covering structure can be determined are constructed, and then used to construct secret sharing schemes with nice access structures.

In this paper, error-correcting codes from perfect nonlinear mappings are constructed, and then employed to construct secret sharing schemes. The error-correcting codes obtained in this paper are very good in general, and many of them are optimal or almost optimal. The secret sharing schemes obtained in this paper have two types of access structures. The first type is democratic in the sense that every participant is involved in the same number of minimal-access sets. In the second type of access structures, there are a few dictators who are in every minimal access set, while each of the remaining participants is in the same number of minimal-access sets.

Constant-composition codes are a special class of constant-weight codes. They include permutation codes as a subclass. In this correspondence, two classes of optimal constant-composition codes are constructed. Both constructions employ perfect nonlinear functions, but they are different.

Constant-composition codes are a special class of constant-weight codes with very strong constraints. It is hard to construct optimal constant-composition codes. There are only a few classes of such optimal codes in the literature. In this correspondence, a family of optimal ternary constant-composition codes is constructed from a class of newly discovered perfect nonlinear functions. This class of codes is related to a new family of skew Hadamard difference sets which are the only examples of such difference sets discovered in the last seventy years.

Constant-composition codes (CCCs) are a special class of constant-weight codes. They include permutation codes as a subclass. In this correspondence, a link between CCCs and generalized double resolvable packing designs is developed, and used to construct several infinite series of optimal CCCs.

Optical orthogonal codes (OOCs) have applications in optical code-division multiple-access communications systems and other wideband code-division multiple environments. They can also be used to construct protocol sequences for multiuser collision channel without feedback, and constant-weight codes for error detection and correction. We have given a cyclotomic construction of several classes of (2/sup m/-1,w,2) OOCs recently. The purpose of this paper is to present five classes of (q-1,w,2) OOCs, and thus five classes of binary constant-weight cyclic codes, where q is a power of an odd prime.

Recently, highly nonlinear functions have been successfully employed to construct authentication codes with and without secrecy. In this paper, we construct four classes of systematic authentication codes from perfect nonlinear functions and almost-perfect nonlinear functions. The systematic authentication codes presented in this paper are either better than existing codes or as good as the best codes known.

A general method of generating families of binary sequences with low correlation as well as large linear span is presented. The lower bound on the linear span is on the order of the square root of the period of each sequence within the family. The design makes use of the theory of function fields. Two example applications of this method are presented in which the underlying function fields are the rational and elliptic function fields respectively.

A permutation array (PA) of length n and minimum distance d is a set of permutations of n elements such that any two permutations coincide in at most n - d positions. Some constructions of PAs are given.

Irreducible cyclic codes have been an interesting subject of study for many years. The weight distribution of some of them have been determined. We determine the minimum distance and certain weights of the duals of binary irreducible cyclic codes. We show that the weight distribution of these codes is determined by the cyclotomic numbers of certain order. As a byproduct, we describe a class of double-error correcting codes.

Almost difference sets have interesting applications in cryptography and coding theory. We give a well-rounded treatment of known families of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period n/spl equiv/0 (mod 4) with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of divisible difference sets (DDSs). We also point out that a result due to Jungnickel (1982) can be used to construct almost difference sets and sequences of period 4l with optimal autocorrelation.

In this correspondence we give several new families of binary sequences of period N with optimal three-level autocorrelation, where N/spl equiv/2 (mod 4). These sequences are either balanced or almost balanced. Our construction is based on cyclotomy.

We describe a class of codes over GF(q), where q is a power of an odd prime. These codes are analogs of the binary Reed-Muller codes and share several features in common with them. We determine the minimum weight and properties of these codes. For a subclass of codes we find the weight distribution and prove that the minimum nonzero weight codewords give 1-designs.

We construct a class of codes of length n such that the minimum distance d outside of a certain subcode is, up to a constant factor, bounded below by the square root of n, a well-known property of quadratic residue codes. The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizations as special cases. We show that most of the special properties of these codes have analogues for split group codes, and present examples of new classes of codes obtained by this construction.

Based on a construction of Xing, Niederreiter, and Lam (see ibid., vol.45, p.2498-2501, 1999), some new linear codes are found from suitable algebraic curves over finite fields. These codes have better parameters compared with Brouwer's table.

In this paper we present a binary-tree approach to the construction of all binary duadic codes of length n = pm . We also calculate the number of binary duadic codes of length n = pm , where p ≡ +1 (mod 8) is a prime.

We first introduce a generalized cyclotomy of order 2 with respect to p/sub 1//sup e(1)/...p/sub t//sup e(t)/. We then present two classes of new error-correcting binary cyclic codes of length p/sub 1//sup e(1)/...p/sub t//sup e(t)/ based on this generalized cyclotomy. We prove either a square-root bound or a similar bound on the minimum odd weight of those codes with length n=p/sub 1//sup e(1)/ p/sub 2//sup e(2)/.

In this correspondence we describe several classes of binary sequences with three-level autocorrelation. Those classes of binary sequences are based on cyclic almost difference sets. Some classes of binary sequences have optimum autocorrelation.

We present a cyclotomic approach to the construction of all binary duadic codes of prime lengths. We calculate the number of all binary duadic codes for a given prime length and that of all duadic codes that are not quadratic residue codes. We give necessary and sufficient conditions for p such that all binary duadic codes of length p are quadratic residue (QR) codes. We also show how to determine some weights of duadic codes with the help of cyclotomic numbers.

Legendre sequences have a number of interesting randomness properties and are closely related with quadratic residue codes. We give lower and upper bounds on the number of patterns distributed in a cycle of the Legendre sequences and establish the relationship between the weight distribution of quadratic residue codes and the pattern distribution of Legendre sequences. Our result shows that Legendre sequences have an ideal distribution of patterns of length s, when s is not large compared with log/sub 2/N, where N is the prime used to define the sequence.

The generalized cyclotomic sequence of order two has several good randomness properties and behaves like the Legendre sequence in several aspects. We calculate the autocorrelation values of the generalized cyclotomic sequence of order two. Our result shows that this sequence could have very good autocorrelation property and pattern distributions of length two if the two primes are chosen properly.

We determine the linear complexity of all Legendre sequences and the (monic) feedback polynomial of the shortest linear feedback shift register that generates such a Legendre sequence. The result shows that Legendre sequences are quite good from the linear complexity viewpoint.