Nearly optimal codebooks based on generalized Jacobi sums

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

Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

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

Several classes of cyclic codes with either optimal three weights or a few weights

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 page

A construction of qq-ary linear codes with two weights

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 page

The Subfield Codes of [q+1,2,q][q+1, 2, q] MDS Codes

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

The Subfield Codes of Hyperoval and Conic codes

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

Near MDS codes from oval polynomials

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 Subfield Codes of Ovoid Codes

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

Minimal Linear Codes over Finite Fields

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

Optimal Binary Linear Codes from Maximal Arcs

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