The Schur product as well as square of liner codes have recently drawn increasing attention from scholars as they are important in both coding theory and cryptography. The objective of this paper is to study the parameters of the products and squares of some families of BCH codes, where BCH codes are an interesting type of cyclic codes. Firstly, we determine the fourth largestq-cyclotomic coset leader moduloqm− 1 denoted as Δ4and then derive the accurate parameters of the corresponding BCH codeC(Δ4). Secondly, letC1andC2be two primitive narrow-sense BCH codes over Fqwith designed distances δaand δb, respectively, where 2 ≤ δa, δb≤n. Form≥ 22 and δa∈ [q⌊m+1/2⌋+1+q⌊m/2⌋−2+2,q⌊m+1/2 ⌋+2 +q⌊m/2⌋−3+1], we give a sufficient and necessary condition forC(δa,δb) ≠ Fnqwhich is closely related to Δ4, whereC(δa,δb) =C1⋆C2is the Schur product ofC1andC2. Thirdly, when δa=q⌊m+1/2⌋+2+q⌊m/2 ⌋−3+1 and Δ4+1 ≤ δb≤ Δ3withm≥ 22, the parameters of C(δa,δb) are studied, where C(δa,δb) is also a primitive narrow-sense BCH code. Fourthly, for a primitive narrow-sense BCH codeCwith designed distance δ, the parameters of its Schur square C2(δ) are investigated when Δ4+1 ≤ δ ≤ Δ3for the third largestq-cyclotomic coset leader Δ3moduloqm− 1. The dimensions of the products and squares of the BCH codes are explicitly determined and lower bounds on their minimum distances are given. Particularly, some codes have good parameters according to the Code Tables at http://www.codetables.de/.

Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let Fpmbe the finite field withpmelements for a primepand a positive integerm. LetV(p)nbe ann-dimensional vector space over Fpfor a primep. In this paper, we study the hybrid character sums of the form Σx∈V(p)nψ (F(x)) χa(x), whereFis a function fromV(p)nto Fpm,a∈V(p)n, ψ is a nontrivial multiplicative character of Fpm, χa(x) = ζ⟨a,x⟩npis the character ofV(p)nand ⟨, ⟩ndenotes a (non-degenerate) inner product ofV(p)n. IfF(x) is a vectorial dual-bent function anda∈V(p)n\ {0}, we determine their complex modulus or explicit values under certain conditions. This generalizes some known results as special cases. We show that the hybrid character sums from vectorial dual-bent functions have very small complex modulus. As applications, three families of asymptotically optimal complex codebooks are constructed from vectorial dual-bent functions and their maximal cross-correlation amplitude are determined based on the hybrid character sums. The codebooks we construct have very small alphabet sizes. This enhances their appeal for implementation. Besides, all of the three families of codebooks have only two-valued or three-valued cross-correlation amplitudes.

In recent years, quasi-complementary sequence sets (QCSSs) have attracted widespread attention as they can support more users in MC-CDMA communications than perfect complementary sequence sets (PCSSs). The objective of this paper is to present three novel constructions of asymptotically optimal or near-optimal periodic QCSSs based on algebraic methods. Firstly, we propose a generic constriction of QCSSs with small alphabet sizepfrom polynomials over finite fields. Using the quadratic and cubic polynomials, we then respectively derive an infinite family of asymptotically optimal QCSSs and an infinite family of asymptotically near-optimal periodic QCSSs with large set sizes. Secondly, we give a construction of periodic QCSSs based on Gaussian sums which have smaller periodic tolerance than that of a known family of QCSSs. Thirdly, we present a construction of periodic QCSSs from permutation polynomials and complementary sets, yielding an infinite family of QCSSs with large set size, small periodic tolerance and low column sequence peak-to-average power ratio (PAPR).

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code division multiple access (MC-CDMA) systems as they can support more users than perfect complementary sequence sets (PCSSs). The objective of this paper is to present new constructions of asymptotically optimal periodic and aperiodic QCSSs with large set sizes. Firstly, we construct a family of asymptotically optimal periodic (p2n,pn− 1,pn− 1,pn+ 1) QCSSs with small alphabet sizep, which has larger set size than the known family of periodic (pn(pn−1),pn−1,pn−1,pn+1) QCSSs. Secondly, we construct five new families of asymptotically optimal aperiodic QCSSs with large set sizes and low aperiodic tolerances. Each family of these aperiodic QCSSs has set size Θ(K2) for some flock sizeK. Compared with known asymptotically optimal aperiodic QCSSs in the literature, our proposed aperiodic QCSSs have better or new parameters. Particularly, for three families of the costructed aperiodic QCSSs, the column sequence peak-to-average power ratio (PAPR) is upper bounded by p if we select suitable column orthogonal complex matrices.

The Rosenbloom-Tsfasman metric (RT-metric for short) is a generalization of the Hamming metric. Matrix codes in the frame of the RT-metric have been used in information transmission over parallel channels. In this paper, we develop some new upper bounds on the minimum RT-distance of an [h×n, k, dRT] linear matrix code, which generalize the Singleton-type bound derived by Rosenbloom and Tsfasman. It should be emphasized that the upper bounds build a connection between the RT-metric and the Hamming metric. Constructions of linear matrix codes are presented and their parameters for the RT-metric are investigated. It is shown that every linear matrix code can be expressed by using the trace function, which is a generalization of the well-known defining-set construction of linear codes. Moreover, we obtain several classes of optimal linear matrix codes in this paper.

Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic. In this paper, based on the augmentation technique, we present two families of linear codes from some functions over finite fields. The first family of linear codes is constructed from monomial functions over finite fields. The weight distribution of the codes is determined in some cases. The codes are proved to be both optimally or almost optimally extendable and self-orthogonal under certain conditions. The localities of the codes and their duals are also studied and we obtain an infinite family of optimal or almost optimal locally recoverable codes. The second family of linear codes is constructed from weakly regular bent functions over finite fields and its weight distribution is explicitly determined. This family of codes is also proved to be both optimally or almost optimally extendable and self-orthogonal. Besides, this family of codes has been proven to have locality 2 or 3 under certain conditions. Particularly, we derive two infinite families of optimal locally recoverable codes. Some infinite families of 2-designs are obtained from the codes in this paper as byproducts.

The self-orthogonality and divisibility are two important properties of linear codes. It is interesting to establish relationship between them. By the well-known Gleason-Pierce-Ward Theorem, all self-dual divisible codes have been totally classified. However, the relationship between the self-orthogonality and divisibility of a q-ary linear codes is known only for$q=2,3$by Huffman and Pless in 2003. It has remained open for more than 20 years to consider other cases. The purpose of this paper is to settle this open problem under certain conditions and construct new families of self-orthogonal codes. Let q be a power of an odd prime p. Firstly, we prove that any p-divisible code containing the all-1 vector over the finite field${\mathbb {F}}_{q}$is self-orthogonal. More generally, it is concluded that any p-divisible$[n,k]$linear code over${\mathbb {F}}_{q}$containing codewords of weight n is monomially equivalent to an$[n,k]$self-orthogonal code over${\mathbb {F}}_{q}$. This result provides a very efficient way to find self-orthogonal codes from p-divisible codes. Secondly, we apply this result to construct self-orthogonal codes with excellent parameters or nice applications. For one thing, we use this result to study the self-orthogonality of generalized Reed-Muller codes, certain projective two-weight codes, and Griesmer codes. For another thing, by this useful result as well as the extending and augmentation techniques for linear codes, we construct eight new families of self-orthogonal divisible codes. These self-orthogonal codes and their duals contain many optimal or almost optimal codes. Besides, some self-orthogonal codes support combinatorial designs and some of them are proved to be optimal or almost optimal locally recoverable codes.

Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime powerq, it is in general difficult to constructq-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a generalq-ary linear code is self-orthogonal or not. In this paper, we mainly present several families ofq-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families ofq-ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families ofq-ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families ofq-ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.

Projective linear codes are a special class of linear codes whose duals have minimum distance at least 3. The columns of the generator matrix of an$[n,k]$projective code over finite field${\mathbb {F}}_{q}$can be viewed as points in the projective space$\text {PG}(k-1, {\mathbb {F}}_{q})$. Projective codes are of interest not only because their duals have good error correcting capability but also because they may be related to interesting combinatorial structures. The objective of this paper is to construct projective linear codes with five families of almost difference sets. To this end, the augmentation and extension techniques for linear codes are used. The parameters and weight distributions of the projective codes are explicitly determined. Several infinite families of optimal or almost optimal codes including MDS codes, near MDS codes, almost MDS odes and Griesmer codes are obtained. Besides, we also give some applications of these codes.

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.

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection.

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.

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.

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.

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.

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega's result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega.