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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. [ABSTRACT FROM AUTHOR]

The puncturing technique is sometimes efficient in constructing projective codes from original codes which are not projective. In this paper, several families of projective linear codes punctured from reducible cyclic codes, special linear codes or irreducible cyclic codes are investigated. The parameters and weight enumerators of the punctured codes and their duals are explicitly determined. Some of the codes are optimal and some of the codes are self-orthogonal which can be used to construct quantum codes. Several infinite families of combinatorial $ 2 $-designs and $ 3 $-designs including some families of Steiner systems are constructed from the punctured codes and their duals. Besides, infinite families of strongly regular graphs are also derived from some families of two-weight projective codes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. 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 $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

The binary Hamming codes with parameters $[{2^{m}-1, 2^{m}-1-m, 3}]$ are perfect. Their extended codes have parameters $[{2^{m}, 2^{m}-1-m, 4}]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[{2^{m+s}+2^{s}-2^{m},2^{m+s}+2^{s}-2^{m}-2m-2,4}]$ , 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 $[{2^{m}+2, 2^{m}-2m, 6}]$. Both classes of binary linear codes have new parameters. [ABSTRACT FROM AUTHOR]

Ovoids in ${\mathrm {PG}}(3, {\mathrm {GF}}(q))$ 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 ${\mathrm {PG}}(3, {\mathrm {GF}}(q))$ corresponds to a $[q^{2}+1, 4, q^{2}-q]$ code over ${\mathrm {GF}}(q)$ , 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. [ABSTRACT FROM AUTHOR]

Abstract Hyperovals in PG (2 , GF (q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG (2 , GF (q)) are equivalent to [ q + 2 , 3 , q ] MDS codes over GF (q) , called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG (2 , GF (q)) for odd q are equivalent to [ q + 1 , 3 , q − 1 ] MDS codes over GF (q) , which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p -ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p -ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p -ary codes seem new. [ABSTRACT FROM AUTHOR]

Abstract As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure two-party computation. Constructing minimal linear codes with new and desirable parameters has been an interesting research topic in coding theory and cryptography. Ashikhmin and Barg showed that w min / w max > (q − 1) / q is a sufficient condition for a linear code over the finite field GF (q) to be minimal, where q is a prime power, w min and w max denote the minimum and maximum nonzero weights in the code, respectively. The first objective of this paper is to present a sufficient and necessary condition for linear codes over finite fields to be minimal. The second objective of this paper is to construct an infinite family of ternary minimal linear codes satisfying w min / w max ≤ 2 / 3. To the best of our knowledge, this is the first infinite family of nonbinary minimal linear codes violating Ashikhmin and Barg's condition. [ABSTRACT FROM AUTHOR]

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 $w_{\min }/w_{\max } \leq 1/2$ from a generic construction, where $w_{\min }$ and $w_{\max }$ , respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined. [ABSTRACT FROM AUTHOR]

Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of q -ary linear codes from trace and norm functions over finite fields. The weight distributions of the linear codes are determined in some cases based on Gauss sums. It is interesting that our construction can produce optimal or almost optimal codes. Furthermore, we show that our codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs with new parameters. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Codebooks (also called signal sets) meeting the Welch bound or the Levenshtein bound are used to distinguish among the signals of different users in CDMA systems. Recently, Zhou, Ding and Li proposed a construction of optimal codebooks based on a set of bent functions satisfying certain conditions over finite fields. In this paper, inspired by the work of Zhou, Ding and Li, we introduce a new construction of codebooks from generalized Boolean bent functions over $\mathbb {Z}_{4}$ . Using this construction, we obtain some optimal codebooks achieving the Levenshtein bound. The codebooks constructed in this paper have parameters (2+2, 2) and a small alphabet size 6. In particular, a set of generalized Boolean bent functions satisfying certain conditions is constructed. As byproducts, some Boolean bent functions are derived. [ABSTRACT FROM AUTHOR]

In this paper, we mainly determine the weight distributions of three classes of linear codes. Firstly, we prove that two classes of ternary linear codes from the following two planar functions have two or three weights: f ( x ) = x 3 k + 1 2 , x ∈ F 3 m , where k , m are odd, gcd ( m , k ) = 1 , and f ( x ) = x 3 k + 1 , x ∈ F 3 m , where m gcd ( m , k ) is odd. They are exactly a part of the open problem in Ding and Ding (2015 Section IV). Secondly, we construct a new class of binary linear codes with three weights. In particular, the linear codes in this paper have applications in consumer electronics, communication and secret sharing schemes. [ABSTRACT FROM AUTHOR]

Projective linear codes are a special class of linear codes whose dual codes have minimum distance at least 3. Projective linear codes with only a few weights are useful in authentication codes, secret sharing schemes, data storage systems and so on. In this paper, two constructions of q -ary linear codes are presented with defining sets given by the intersection and difference of two sets. These constructions produce several families of new projective two-weight or three-weight linear codes. As applications, our projective codes can be used to construct secret sharing schemes with interesting access structures, strongly regular graphs and association schemes with three classes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Two-weight linear codes have many wide applications in authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we present two classes of two-weight binary or ternary linear codes. In some cases, they are optimal or almost optimal. They can also be used to construct secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a generalized Gray map G on Z p k with p a prime, which generalizes a Carlet's result in [1] . For a Z p k -valued function f ( x ) , we use exponential sums to express the Hamming weight of G ( f ( x ) ) . As an application, a family of nonlinear codes over F p is obtained from the generalized Gray map. We use the Weil-type exponential sums over Galois rings to provide a lower bound for the minimum distance of these codes. [ABSTRACT FROM AUTHOR]

In this letter, we use Gauss sums of index 2 to construct a class of new binary linear codes with at most three weights. In special cases, they are optimal or near optimal codes. These codes can be used in authentication codes and secret sharing schemes. [ABSTRACT FROM PUBLISHER]

A linear code with parameters of the form [ n , k , n − k + 1 ] is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form [ n , k , n − k ] is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost maximum distance separable. Near MDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There are many unsolved problems about near MDS codes. It is hard to construct an infinite family of near MDS codes whose weight distributions can be settled. In this paper, seven infinite families of [ 2 m + 1 , 3 , 2 m − 2 ] near MDS codes over GF (2 m) and seven infinite families of [ 2 m + 2 , 3 , 2 m − 1 ] near MDS codes over GF (2 m) are constructed with special oval polynomials for odd m. In addition, nine infinite families of optimal [ 2 m + 3 , 3 , 2 m ] near MDS codes over GF (2 m) are constructed with oval polynomials in general. The weight distributions of these near MDS codes are settled. [ABSTRACT FROM AUTHOR]