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Linear codes are widely studied due to their applications in communication, cryptography, quantum codes, distributed storage and many other fields. In this paper, we use the trace and norm functions over finite fields to construct a family of linear codes. The weight distributions of the codes are determined in three cases via Gaussian sums. The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases. In particular, we prove that this family of linear codes has locality 2. Several optimal or almost optimal linear codes and locally recoverable codes are derived. In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained. The self-orthogonal codes derived in this paper can be used to construct lattices and have nice application in distributed storage., Comment: 25 pages

In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums to determine the parameters and weight distributions of the augmented codes in some cases. It is shown that the augmented codes are self-orthogonal and have only a few nonzero weights. For another thing, the locality of the augmented codes is proved to be 2, which indicates the augmented codes are useful in distributed storage. Besides, the augmented codes are projective as the minimum distance of their duals is proved to be 3. In particular, we obtain several (almost) optimal linear codes and locally recoverable codes., Comment: 25 pages

MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three., Comment: 22 pages

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 locality of them is determined and the weight distributions of two subfamilies of the codes are also given. An infinite family of locally recoverable codes which are at least almost optimal and some optimal recoverable codes are obtained from the linear codes. In particular, the two subfamilies of the codes are proved to be both optimally or almost optimally extendable and self-orthogonal. The second family of linear codes is constructed from weakly regular bent functions over finite fields and their weight distribution is determined. This family of codes is proved to have locality 3 for some cases and is conjectured to have locality 2 for other cases. Particularly, two families of optimal locally recoverable codes are derived from the linear codes. Besides, this family of codes is also proved to be both optimally or almost optimally extendable and self-orthogonal., Comment: 44 pages

Self-orthogonal codes are an important subclass of linear codes which have nice applications in quantum codes and lattices. It is known that a binary linear code is self-orthogonal if its every codeword has weight divisible by four, and a ternary linear code is self-orthogonal if and only if its every codeword has weight divisible by three. It remains open for a long time to establish the relationship between the self-orthogonality of a general $q$-ary linear code and the divisibility of its weights, where $q=p^m$ for a prime $p$. In this paper, we mainly prove that any $p$-divisible code containing the all-1 vector over the finite field $\mathbb{F}_q$ is self-orthogonal for odd prime $p$, which solves this open problem under certain conditions. Thanks to this result, we characterize that any projective two-weight code containing the all-1 codeword over $\mathbb{F}_q$ is self-orthogonal. Furthermore, by the extending and augmentation techniques, we construct six new families of self-orthogonal divisible codes from known cyclic codes. Finally, we construct two more families of self-orthogonal divisible codes with locality 2 which have nice application in distributed storage systems., Comment: 61 pages

In ``Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428'', Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding $2$-designs or $3$-designs. Up to now, there are only a few known infinite families of NMDS codes holding $t$-designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding $t$-designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold $2$-designs or $3$-designs. Compared with known $t$-designs from NMDS codes, ours have different parameters. Besides, several infinite families of optimal locally recoverable codes are also derived via the NMDS codes., Comment: 34 pages

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold $t$-designs, where $t=2,3$. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived., Comment: 21 pages

Locally recoverable codes are widely used in distributed and cloud storage systems. The objective of this paper is to present a construction of near MDS codes with oval polynomials and then determine the locality of the codes. It turns out that the near MDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes. The lengths of the locally recoverable codes are different from known ones in the literature., Comment: arXiv admin note: substantial text overlap with arXiv:2204.11208

A linear code with parameters $[n,k,n-k]$ is said to be almost maximum distance separable (AMDS for short). An AMDS code whose dual is also AMDS is referred to as an near maximum distance separable (NMDS for short) code. NMDS codes have nice applications in finite geometry, combinatorics, cryptography and data storage. In this paper, we first present several constructions of NMDS codes and determine their weight enumerators. In particular, some constructions produce NMDS codes with the same parameters but different weight enumerators. Then we determine the locality of the NMDS codes and obtain many families of distance-optimal and dimension-optimal locally repairable codes.

A class of binary sequences with period $2p$ is constructed using generalized cyclotomic classes, and their linear complexity, minimal polynomial over ${\mathbb{F}_{{q}}}$ as well as 2-adic complexity are determined using Gauss period and group ring theory. The results show that the linear complexity of these sequences attains the maximum when $p\equiv \pm 1(\bmod~8)$ and is equal to {$p$+1} when $p\equiv \pm 3(\bmod~8)$ over extension field. Moreover, the 2-adic complexity of these sequences is maximum. According to Berlekamp-Massey(B-M) algorithm and the rational approximation algorithm(RAA), these sequences have quite good cryptographyic properties in the aspect of linear complexity and 2-adic complexity., Comment: 23 pages

Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of $q$-ary projective two-weight linear codes is presented, where $q$ is a power of 2. The parameters of both the codes and their duals are excellent. As applications, the codes are used to derive strongly regular graphs with new parameters and secret sharing schemes with interesting access structures.

Recently, subfield codes of geometric codes over large finite fields $\gf(q)$ with dimension $3$ and $4$ were studied and distance-optimal subfield codes over $\gf(p)$ 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 $\gf(q)$, and then studies the subfield codes over $\gf(p)$ of some of the $[q+1, 2,q]$ MDS codes over $\gf(q)$. Two families of dimension-optimal codes over $\gf(p)$ are obtained, and several families of nearly optimal codes over $\gf(p)$ are produced. Several open problems are also proposed in this paper.

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. 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 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.

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.

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 quadratics and the second is the Tits ovoids. It is known that an ovoid in $\PG(3, \gf(q))$ corresponds to a $[q^2+1, 4, q^2-q]$ code over $\gf(q)$, which is called an ovoid code. The objectives of this paper is 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.

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 $w_{\min}/w_{\max}\leq 2/3$. To the best of our knowledge, this is the first infinite family of nonbinary minimal linear codes violating Ashikhmin and Barg's condition.

Codebooks with small inner-product correlation are applied in many practical applications including direct spread code division multiple access (CDMA) communications, space-time codes and compressed sensing. It is extremely difficult to construct codebooks achieving the Welch bound or the Levenshtein bound. Constructing nearly optimal codebooks such that the ratio of its maximum cross-correlation amplitude to the corresponding bound approaches 1 is also an interesting research topic. In this paper, we firstly study a family of interesting character sums called generalized Jacobi sums over finite fields. Then we apply the generalized Jacobi sums and their related character sums to obtain two infinite classes of nearly optimal codebooks with respect to the Welch or Levenshtein bound. The codebooks can be viewed as generalizations of some known ones and contain new ones with very flexible parameters.

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., Comment: 19 pages

Recently, linear codes with a few weights were widely investigated due to their applications in secret sharing schemes and authentication schemes. In this letter, we present a class of $q$-ary linear codes derived from irreducible cyclic codes with $q$ a prime power. We use Gauss sums to represent its Hamming weights and obtain the bounds of its minimum Hamming distance. In some cases, we explicitly determine its weight distributions which generalize some known results. It is quite interesting that many new codes obtained in this letter are optimal according to some bounds on linear codes., Comment: This paper has been withdraw because a part of the main results has been reported before

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.

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 a general construction of cyclic codes. As applications, we obtain a class of optimal three-weight codes achieving the Griesmer bound, which generalizes a Vega's result in \cite{V1}, and several classes of cyclic codes with only a few weights, which solve the open problem in \cite{V1}., Comment: 24 pages