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In this paper, we give a new method to answer a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the equation x d + (x + 1) d = b in F q 4 , where n is a positive integer, q = 2 n and d = q 3 + q 2 + q − 1. In particular, we directly determine the differential spectrum of this power function x d using a more simple and direct method compared with those in the literature. [ABSTRACT FROM AUTHOR]

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7]. [ABSTRACT FROM AUTHOR]

The symbol-pair read channel was first proposed by Cassuto and Blaum. Later, Yaakobi et al. generalized it to the b -symbol read channel. It is motivated by the limitations of the reading process in high density data storage systems. One main task in b -symbol coding theory is to determine the b -symbol weight hierarchy of codes. In this paper, we study the b -symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes. The complete symbol-pair weight distribution of the Kasami codes is determined. [ABSTRACT FROM AUTHOR]

Recently, several infinite families of minimal and optimal linear codes are obtained from simplicial complexes or down-sets by Hyun and Wu et al. Motivated by their work, we give two new constructions of codes over the ring F p + u F p , u 2 = u by using down-sets. We compute the Lee weight distributions of two classes of codes when the down-sets are generated by a single maximal element. Moreover, by using the Gray map, we obtain two classes of codes, which are minimal and distance optimal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Toeplitz codes of index t , which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1 t over F q , so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [ 32 , 16 , 8 ] , [ 33 , 11 , 11 ] and [ 34 , 17 , 8 ] codes with improved parameters directly. [ABSTRACT FROM AUTHOR]

We study codes with parameters of the ternary Hamming (n = (3 m − 1) / 2 , 3 n − m , 3) code, i.e., ternary 1-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary 1-perfect codes of rank n − m + 1 , count their number, and prove that all such codes can be obtained from each other by a sequence of two-coordinate switchings. We enumerate ternary 1-perfect codes of length 13 obtained by concatenation from codes of lengths 9 and 4; we find that there are 93241327 equivalence classes of such codes. [ABSTRACT FROM AUTHOR]

In the paper, we firstly study the algebraic structures of Z p Z p k -additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for the Z p Z p 2 -additive cyclic code whose Gray image is linear (not necessarily cyclic) over Z p is given. Moreover, as for some special families of cyclic codes over Z p 2 and Z p Z p 2 , the linearity of the Gray images is determined. [ABSTRACT FROM AUTHOR]

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring F p + u F p + v F p + u v F p , where u 2 = 0 , v 2 = 0 , u v = v u . These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over F p . In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given. [ABSTRACT FROM AUTHOR]

We construct an infinite family of two-Lee-weight codes over the ring F 2 + u F 2 + v F 2 + u v F 2 . These codes are defined as trace codes and have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Then, taking Gray mapping, we obtain an infinite family of abelian binary two-weight codes which are shown to be optimal by application of the Griesmer bound. Moreover, an application to secret sharing schemes is given. [ABSTRACT FROM AUTHOR]

We show that no more new distance-regular graphs in the tables of the book of (Brouwer, Cohen, Neumaier, 1989) can be produced by using the coset graph of additive completely regular codes over finite fields. [ABSTRACT FROM AUTHOR]

This paper is devoted to presenting two counterexamples to Shiromoto's (1999) results [2] on MacWilliams type identities with respect to Lee weight enumerators and Euclidean weight enumerators over Z ℓ . [ABSTRACT FROM AUTHOR]

Linear codes over finite rings are described here as trace codes for a suitable generalization of the trace called a GF-trace. Cyclic codes over Galois rings are given a trace description as well. The main tools are the notion of trace dual bases, in the case of linear codes, and of normal bases of an extension ring over a ring, in the case of cyclic codes. [ABSTRACT FROM AUTHOR]