A linear code with parameters [n, k, n–k+ 1] is called maximum distance separable (MDS), and one with parameters [n; k; n–k] is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorialt-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions fort> 2, all with length at mostq+ 1. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater thanq+ 1 are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the firstgeneric constructionofq-ary NMDS codes supporting 2-designs with lengthsexceedingq+1. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

Gold functions are well-known for designing sequences with good correlation properties. This paper reveals a fascinating relationship between Gold functions and Golay-Davis-Jedwab (GDJ) Boolean functions to design sequences with low correlation and low PAPR, which is the first of its kind. The bent and semi-bent properties of the Gold functions are utilized to derive the correlation of the resultant binary spreading sequence sets, while the complementary properties of the GDJ Boolean functions are used to derive the low PAPR. Based on this idea we propose three new sets of binary spreading sequences with low PAPR and low correlation. The proposed sequence sets have the potential to be used as spreading sequences in non-orthogonal multiple access (NOMA). They have a very large set size, which in turn results in a high overloading factor as compared to the existing sequence sets with the same length and the same correlation, which are generated through systematic constructions.

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.

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.

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

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.

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented.

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.

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of Fqto construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of Fq, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes.

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.

Linear complementary dual (LCD) cyclic codes were referred historically 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 particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results on the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.

Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against sidechannel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with o complementary dual (σ-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, σ-LCD codes can also be used to construct LCP of codes. We show that for q 2, all q-ary linear codes are σ-LCD, and for every binary linear code C, the code {0} × C is σ-LCD. Furthermore, we study deeply σ-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of σ-LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good σ-LCD GQC codes. Finally, we focus on σ-LCD abelian codes and prove that all abelian codes in a semisimple group algebra are σ-LCD. The results derived in this paper extend those on the classical LCD codes and show that σ-LCD codes allow the construction of LCP of codes more easily and with more flexibility.

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.

Linear codes with complementary duals (LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Nonbinary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over Fq(q > 3) is equivalent to a Euclidean LCD code and any linear code over Fq2(q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over Fqwith q > 3 exists if there is an [n, k, d]-linear code over Fqand an [n, k, d]-linear Hermitian LCD code over Fq2with q > 2 exists if there is an [n, k, d]-linear code over Fq2. Hence, when q > 3 (resp. q > 2) q-ary Euclidean (resp. q2-ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q2-ary linear codes). This gives a direct proof that every triple of parameters [n, k, d] which is attainable by linear codes over Fqwith q > 3 (resp. over Fq2with q > 2) is attainable by Euclidean LCD codes (resp. by Hermitian LCD codes). In particular there exist families of q-ary Euclidean LCD codes (q > 3) and q2-ary Hermitian LCD codes (q > 2) exceeding the asymptotical Gilbert-Varshamov bound. Further, we give a second proof of these results using the theory of Gröbner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.

Linear complementary dual (LCD) codes are a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called sidechannel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from any algebraic curve. Then, some explicit LCD codes from elliptic curves are presented. Maximum distance separable (MDS) codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we also derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.

Let p be an odd prime. Recently, Camarero and Martínez (in “Quasi-perfect Lee codes of radius 2 and arbitrarily large dimension”, IEEE Trans. Inform. Theory, vol. 62, no. 3, 2016) constructed some p-ary 2-quasi-perfect Lee codes for p ≡ ±5 (mod 12). In this paper, some infinite classes of p-ary 2-quasi-perfect Lee codes for any odd prime p with flexible length and dimension are presented. More specifically, we provide a new method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to obtain two classes of 2-quasi-perfect Lee codes defined in the space Zpnfor n = pk+1/2 (with p ≡ 1, -5 (mod 12) and k is any integer, or p ≡ -1, 5 (mod 12) and k is an even integer) and n = pk-1/2 (with p ≡ -1, 5 (mod 12), k is an odd integer and pk> 12). Our codes encompass the p-ary (p ≡ ±5 (mod 12)) 2-quasiperfect Lee codes constructed by Camarero and Martínez. Furthermore, we prove that the related Cayley graphs are Ramanujan or almost Ramanujan using Kloosterman sums. This generalizes the work of Bibak, Kapron, and Srinivasan (in “The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs”, IEEE Trans. Inform. Theory, vol. 62, no. 11, 2016) from the case p ≡ 3 (mod 4) and k = 1 to the case of any odd prime p and positive integer k. Finally, we derive some necessary conditions with the exponential sums of all 2-perfect codes and 2-quasi-perfect codes, and present a heuristic algorithm for constructing 2-perfect codes and 2-quasi-perfect codes. Our results show that, in general, the Cayley graphs associated with 2-perfect codes are Ramanujan. From the algorithm, some new 2-quasi-perfect Lee codes different from those constructed from quadratic curves are given. The Lee codes presented in this paper have applications in constrained and partial-response channels, flash memories, and decision diagrams.

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon's $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh-Hadamard transform of their component functions, and we generalize these results to the case of odd prime.

Plateaued functions are very important cryptographic functions due to their various desirable cryptographic characteristics. We point out that plateaued functions are more general than bent functions (that is, functions with maximum nonlinearity). Some Boolean plateaued functions have large nonlinearity, which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers, and contributes, when they are the component functions of the substitution boxes in block ciphers, to protection against linear cryptanalysis. P-ary plateaued functions have attracted recently some attention in the literature, and many activities on generalized p-ary functions have been carried out. This paper increases our knowledge on plateaued functions in the general context of generalized p-ary functions. We first introduce two new versions of plateaued functions, which we shall call generalized plateaued functions and admissible plateaued functions. The generalized plateaued functions extend the standard notion of plateaued p-ary functions to those whose outputs are in the ring Zpk. Next, we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components. Finally, we provide for the first time two constructions of generalized plateaued functions. In particular, we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes.

In this paper, we investigate properties of generalized bent Boolean functions and 2k-bent (i.e., negabent, octabent, hexadecabent, et al.) Boolean functions in a uniform framework. From the Hadamard matrices, Hodzic and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodzic and Pasalic's conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semibent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and 2k-bent functions, demonstrate that 2k-bent functions are actually a special class of generalized bent functions, and completely characterize 2k-bent functions.

As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu et al. as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over F22mof any algebraic degree between 2 and m. This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager.

In this paper, we present a construction of linear codes over F2tfrom Boolean functions, which is a generalization of Ding's method. Based on this construction, we give two classes of linear codes C̃fand Cfover F2tfrom a Boolean function f : Fq→ F2, where q = 2nand F2tis some subfield of Fq. The complete weight enumerator of C̃fcan be easily determined from the Walsh spectrum of f , while the weight distribution of the code Cfcan also be easily settled. Particularly, the number of nonzero weights of C̃fand C f is the same as the number of distinct Walsh values of f. As applications of this construction, we show several series of linear codes over F2twith two or three weights by using bent, semibent, monomial and quadratic Boolean function f.

Maximum distance separable (MDS) codes introduce MDS matrices which not only have applications in coding theory but also are of great importance in the design of block ciphers. It has received a great amount of attention. In this paper, we first introduce a special generalization of circulant matrices called block circulants with circulant blocks, which can be used to construct MDS matrices. Then we investigate some interesting and useful properties of this class of matrices and prove that their inverse matrices can be implemented efficiently. Furthermore, we present some 4 × 4 and 8 × 8 efficient MDS matrices of this class which are suitable for MDS diffusion layer. Compared with previous results, our construction provides better efficiency for the implementation of both the matrix and the its inverse matrix.

Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. and Zhou et al. to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves an open problem proposed by Ding et al. Furthermore, this paper constructs new linear codes with two or three weights and presents their weight distributions. They contain some optimal codes meeting certain bound on linear codes.

Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes have been studied for many years and find many applications. The determination of the weight distributions of such codes is an open problem. Generally, the weight distributions of cyclic codes are difficult to determine. Utilizing a class of elliptic curves, this paper determines the weight distributions of dual codes ofq-ary cyclic codes with two zeros for a few more cases, whereqis an odd prime power.