Maosheng Xiong

The weight distributions of a class of cyclic codes

Abstract 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.

Weight Distribution of a Class of Cyclic Codes With Arbitrary Number of Zeros

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined.

The weight distributions of a class of cyclic codes II

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) 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. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.

Three New Families of Zero-Difference Balanced Functions With Applications

Zero-difference balanced (ZDB) functions integrate a number of subjects in combinatorics and algebra, and have many applications in coding theory, cryptography, and communications engineering. In this paper, three new families of ZDB functions are presented. The first construction gives ZDB functions defined on the abelian groups (GF(q1)×,...,×GF(qk),+) with new and flexible parameters. The other two constructions are based on 2-cyclotomic cosets and yield ZDB functions on \BBZn with new parameters. The parameters of optimal constant composition codes, optimal, and perfect difference systems of sets obtained from these new families of ZDB functions are also summarized.

Pseudo-cyclic Codes and the Construction of Quantum MDS Codes

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes.

Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279–3286, 2010; IEEE Trans Inf Theory 60:3903–3912, 2014), we study two families of cyclic codes over

The Weight Hierarchy of Some Reducible Cyclic Codes

{\mathbb F}_p

Selmer groups and Tate–Shafarevich groups for the congruent number problem

Fp with arbitrary number of zeroes of generalized Niho type, more precisely