一些不可约循环码的权重分布

令n 为q^m -1 的正因子. 本文主要借助特征标、分圆类及Gauss 周期的知识确定了GF(q) 上码长为n、维数为m 的不可约循环码在一些特殊情形下的权重分布, 这些不可约循环码为二权、三权或是四权码.     

The weight distributions of some cyclic codes with three or four nonzeros over \mathbb F _3

Because of efficient encoding and decoding algorithms comparing with linear block codes, cyclic codes form an important family and have applications in communications and storage systems. However, their weight distributions are known only for a few cases mainly on the codes with no more than three nonzeros. In this paper, the weight distributions of two classes of cyclic codes with three or four nonzeros are determined.     

The weight distributions of two classes of p-ary cyclic codes with few weights

Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p, we investigate two classes of p-ary cyclic codes for special cases and determine their weight distributions explicitly. The results show that both codes have at most five nonzero weights.     

A class of binary cyclic codes with generalized Niho exponents

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.     

All Two-Weight Irreducible Cyclic Codes?

The aim of this paper is the classification of two-weight irreducible cyclic codes. Using Fourier transforms and Gauss sums, we obtain necessary and sufficient numerical conditions for an irreducible cyclic code to have at most two weights. This gives a unified explanation for all two-weight irreducible cyclic codes and allows a conjecturally complete classification. Aside from the two known infinite families of two-weight irreducible cyclic codes, a computer search reveals 11 sporadic examples. We conjecture that these are already all two-weight irreducible cyclic codes and give a partial proof of our conjecture conditionally on GRH.     

Hamming weights in irreducible cyclic codes

The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes.     

Infinite families of 2‐designs from a class of cyclic codes

Combinatorial $t$‐designs have wide applications in coding theory, cryptography, communications, and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$‐design. In this paper, we first determine the weight distributions of a class of linear codes derived from the dual of some extended cyclic codes. We then obtain infinite families of 2‐designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By a simple counting argument, we obtain exponentially many 2‐designs.     

Codes over an infinite family of rings with a Gray map

Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.     

The weight distributions of a class of non-primitive cyclic codes with two nonzeros

Recently, a class of non-primitive cyclic codes with two nonzeros have received much attention of researchers and their weight distributions have been obtained for several cases of two key parameters related to the nonzeros. In this paper, by evaluating certain Jacobi sums, we determine the weight distributions of this class of cyclic codes for one more special case.     

Weights in codes and genus 2 curves

We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry and an application of Weil's theorem. We relate each weight appearing in the dual codes to the number of rational points on a genus 2 curve of 2-rank 1 over a finite field of characteristic 2. The possible values for the number of points on a curve of genus 2 and 2-rank 1 are determined, thus determining the weights in the dual codes.

     

On the Cyclicity of Goppa Codes,Parity-Check Subcodes of Goppa Codes,and Extended Goppa Codes

Classical Goppa codes are a special case of Alternant codes. First we prove that the parity-check subcodes of Goppa codes and the extended Goppa codes are both Alternant codes. Before this paper, all known cyclic Goppa codes were some particular BCH codes. Many families of Goppa codes with a cyclic extension have been found. All these cyclic codes are in fact Alternant codes associated to a cyclic Generalized Reed–Solomon code. In (1989, J. Combin. Theory Ser. A 51, 205–220) H. Stichtenoth determined all cyclic extended Goppa codes with this property. In a recent paper (T. P. Berger, 1999, in “Finite Fields: Theory, Applications and Algorithms (R. Mullin and G. Mullen, Eds.), pp. 143–154, Amer. Math. Soc., Providence), we used some semi-linear transformations on GRS codes to construct cyclic Alternant codes that are not associated to cyclic GRS codes. In this paper, we use these results to construct cyclic Goppa codes that are not BCH codes, new families of Goppa codes with a cyclic extension, and some families of non-cyclic Goppa codes with a cyclic parity-check subcode.     

The weight distributions of a class of cyclic codes

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) [14], Ding et al. (2011) [5], Wang et al. (2011) [20]. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method.     

Two-weight or three-weight binary linear codes from cyclotomic mappings

Recently, linear codes with few weights have been studied extensively. These linear codes have wide applications in secret sharing schemes and authentication codes. In this paper, we introduce a new construction of defining sets using cyclotomic mappings and construct three new classes of binary linear codes with two or three weights. We also explicitly determine the weight distributions of these codes.     

On row-cyclic codes with algebraic structure

In this article, some row-cyclic error-correcting codes are shown to be ideals in group rings in which the underlying group is metacyclic. For a given underlying group, several nonequivalent codes with this structure may be generated. Each is related to a cyclic code generated in response, to the metrics associated with the underlying metacyclic group. Such codes in the same group ring are isomorphic as vector spaces but may vary greatly in weight distributions and so are nonequivalent. If the associated cyclic code is irreducible, examining the structure of its isomorphic finite field yields all nonequivalent codes with the desired structure. Several such codes have been found to have minimum distances equalling those of the best known linear codes of the same length and dimension.This work was presented in part at the First International Conference on Finite Fields, Coding Theory, and Advances in Communications and Computing, Las Vegas, Nevada, August, 1991.     

The edge-forwarding index of orbital regular graphs

We define a graph as orbital regular if there is a subgroup of its automorphism group that acts regularly on the set of edges of the graph as well as on all its orbits of ordered pairs of distinct vertices of the graph. For these graphs there is an explicit formula for the edge-forwarding index, an important traffic parameter for routing in interconnection networks. Using the arithmetic properties of finite fields we construct infinite families of graphs with low edge-forwarding properties. In particular, the edge-forwarding index of Paley graphs is determined. A connection with the Waring problem over finite fields and the coset weight enumeration of certain cyclic codes is established.     

Improvements on Generalized Hamming Weights of Some Trace Codes

We obtain improved bounds for the generalized Hamming weights of some trace codes which include a large class of cyclic codes over any finite field. In particular, we improve the corresponding bounds of Stichtenoth and Voss [8] using various methods altogether.     

环F_2+uF_2上长度为2n(n为奇数)的循环码

研究了环F2+uF2上长度为2n(n为奇数)的循环码,给出了循环码及其对偶码的生成多项式,以及循环码为自对偶码的充要条件,最后进一步给出了循环码极小Lee重量的一些相关结论     

Linear codes from vectorial Boolean power functions

In this paper, three classes of binary linear codes with few weights are proposed from vectorial Boolean power functions, and their weight distributions are completely determined by solving certain equations over finite fields. In particular, a class of simplex codes and a class of first-order Reed-Muller codes can be obtained from our construction by taking the identity map, whose dual codes are Hamming codes and extended Hamming codes, respectively.     

环F2＋uF2上长度为2n（n为奇数）的循环码

研究了环F2＋uF2上长度为2n（n为奇数）的循环码,给出了循环码及其对偶码的生成多项式,以及循环码为自对偶码的充要条件,最后进一步给出了循环码极小Lee重量的一些相关结论     

Strongly regular graphs from reducible cyclic codes

Journal of Algebraic Combinatorics - Let p be a prime number. Reducible cyclic codes of rank 2 over $$\mathbb {Z}_{p^m}$$ are shown to have exactly two Hamming weights in some cases. Their weight...