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

Linear codes with few weights over F2+uF2\mathbb{F}_2+u\mathbb{F}_2

In this paper, we construct an infinite family of five-weight codes from trace codes over the ring $R=\mathbb{F}_2+u\mathbb{F}_2$, where $u^2=0.$ The trace codes have the algebraic structure of abelian codes. Their Lee weight is computed by using character sums. Combined with Pless power moments and Newton's Identities, the weight distribution of the Gray image of trace codes was present. Their support structure is determined. An application to secret sharing schemes is given.Comment: 14 pages, need help in page 12. arXiv admin note: text overlap with arXiv:1612.0096

Index bounds for character sums with polynomials over finite fields

We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings of small indices. As an application, we also give some general bounds for numbers of solutions of some Artin-Schreier equations and mininum weights of some cyclic codes

Deep Holes in Reed-Solomon Codes Based on Dickson Polynomials

For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of $\mathcal{C}$ is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear with a special uniformity. To find families of received words that are not deep holes, we reduce to a subset sum problem (or equivalently, a Dickson polynomial-variation of Waring's problem) and find solution conditions by applying an argument using estimates on character sums indexed over the evaluation set

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

Codes Associated with Orthogonal Groups and Power Moments of Kloosterman Sums

In this paper, we construct three binary linear codes $C(SO^{-}(2,q))$, $C(O^{-}(2,q))$, $C(SO^{-}(4,q))$, respectively associated with the orthogonal groups $SO^{-}(2,q)$, $O^{-}(2,q)$, $SO^{-}(4,q)$, with $q$ powers of two. Then we obtain recursive formulas for the power moments of Kloosterman and 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups. We emphasize that, when the recursive formulas for the power moments of Kloosterman sums are compared, the present one is computationally more effective than the previous one constructed from the special linear group $SL(2,q)$. We illustrate our results with some examples

Two classes of linear codes with a few weights based on twisted Kloosterman sums

Linear codes with a few weights have wide applications in information security, data storage systems, consuming electronics and communication systems. Construction of the linear codes with a few weights and determination of their parameters are an important research topic in coding theory. In this paper, we construct two classes of linear codes with a few weights and determine their complete weight enumerators based on twisted Kloosterman sums

Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments

In this paper, we construct two ternary linear codes associated with the symplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then we obtain recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of "Gauss sums" for the symplectic groups Sp(2n,q).Comment: No comment

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

Codes Associated with O(3,2r)O(3,2^r) and Power Moments of Kloosterman Sums with Trace One Arguments

We construct a binary linear code $C(O(3,q))$, associated with the orthogonal group $O(3,q)$. Here $q$ is a power of two. Then we obtain a recursive formula for the odd power moments of Kloosterman sums with trace one arguments in terms of the frequencies of weights in the codes $C(O(3,q))$ and $C(Sp(2,q))$. This is done via Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups