Search

Let F q be a finite field with q elements, where q is a power of a prime p. In this paper, we obtain an improvement on Weil bounds for character sums associated to a polynomial f(x) over F q , which extends the results of Wan et al. (Des. Codes Cryptogr. 81, 459–468, 2016) and Wu et al. (Des. Codes Cryptogr. 90, 2813–2821, 2022). [ABSTRACT FROM AUTHOR]

Self-orthogonal codes are linear codes such that they are contained in their duals. Self-orthogonal codes have attracted much attention due to their applications in linear complementary dual codes (LCD codes for short), quantum codes, row-self-orthogonal matrices and so on. In this paper, we first construct several families of ternary self-orthogonal codes from weakly regular bent functions. The parameters and weight distributions of them are determined. Then we use the self-orthogonal codes to construct new infinite families of ternary LCD codes. Some LCD codes are optimal according to the Code Tables at http://www.codetables.de/. [ABSTRACT FROM AUTHOR]

Locally recoverable codes are widely used in distributed and cloud storage systems. In this paper, we first 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. [ABSTRACT FROM AUTHOR]

A class of binary sequences with period 2p is constructed using generalized cyclotomic classes, and their linear complexity, minimal polynomial over 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 ≡ ± 1 (mod 8) and is equal to p+1 when p ≡ ± 3 (mod 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 cryptographic properties in the aspect of linear complexity and 2-adic complexity. [ABSTRACT FROM AUTHOR]

Linear codes with a few weights have nice applications in communication, secret sharing schemes, authentication codes, association schemes, block designs and so on. Projective binary linear codes are one of the most important subclasses of linear codes for practical applications. The objective of this paper is to construct projective binary linear codes with some special Boolean functions. Four families of binary linear codes with three or four weights are derived and the parameters of their duals are also determined. It turns out that the duals of these codes are optimal or almost optimal with respect to the sphere-packing bound. As applications, the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures. [ABSTRACT FROM AUTHOR]

Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of q-ary two-weight linear codes including two subfamilies of projective codes is presented, where q is a prime power. 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. [ABSTRACT FROM AUTHOR]

Linear codes are an important class of error-correcting codes and widely used in secret sharing schemes, combinational designs, authentication codes and so on. The objective of this paper is to construct q-ary linear codes with good parameters from irreducible cyclic codes. Several classes of linear codes with a few weights including infinite families of distance-optimal ones are derived. The nonzero codewords of some of the codes in this paper have minimal support for inclusion and such codes can be used to construct secret sharing schemes with interesting access structures. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Complete weight distribution can be used to study authentication codes and the Walsh transform of monomial functions over finite fields. Also, the Hamming weight distribution of a code can be obtained from its complete weight distribution. In this paper, we investigate the complete weight distributions of two classes of cyclic codes. We explicitly present the complete weight enumerators of the cyclic codes. Particularly, we partly solve an open problem proposed in Luo and Feng (IEEE Trans. Inf. Theory 54(12), 5345-5353 (2008)). [ABSTRACT FROM AUTHOR]

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]