Cyclic codes from two-prime generalized cyclotomic sequences of order 6

Tongjiang Yan,Yanyan Liu,Yuhua Sun

doi:10.3934/amc.2016036

Abstract

Cyclic codes have wide applications in data storage systems and communication systems. Employing binary two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field $\mathrm{GF}(q)$ and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds on the minimum distance for some other cyclic codes.

Highlights

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms
Li presented several classes of cyclic codes using another class of two-prime Whiteman generalized cyclotomic sequences of order 4, and gave lower bounds on the nonzero minimum hamming weight of these cyclic codes [16]
We consider the cyclic codes over the finite field of order q GF(q) from another class of two-prime Whiteman generalized cyclotomic sequences of order 6, and present the generator polynomials of these cyclic codes

Summary

Introduction

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. Li presented several classes of cyclic codes using another class of two-prime Whiteman generalized cyclotomic sequences of order 4, and gave lower bounds on the nonzero minimum hamming weight of these cyclic codes [16]. The two-prime sequence is employed to construct several classes of cyclic codes of order d(d > 0) over GF(q) and lower bounds on the minimum weight of these cyclic codes were developed in [7]. The cyclic code Cs generated by the minimal polynomial of sn is called the cyclic code defined by the sequence sn, which is a linear [n, k] code with k = deg(gcd(xn −1, Sn(x))). The generator polynomial g(x) of cyclic code Cs is equal to the minimal polynomial of the sequence sn, i.e., g(x) xn − 1 gcd(xn − 1, S n(x)). The polynomial S(x) in Equation (3) is exactly the generator polynomial of the sequence sn [4]

Generator polynomials of cyclic codes

The minimal distance of the cyclic codes

Conclusion