Cyclic codes from the first class two-prime Whiteman's generalized cyclotomic sequence with order 6

TL;DR

This paper analyzes the autocorrelation and linear complexity of a specific cyclotomic sequence of order 6, and uses it to construct cyclic codes with good properties and bounds on their minimum distance.

Contribution

It introduces the first class two-prime Whiteman's generalized cyclotomic sequence of order 6 and applies it to cyclic code construction.

Findings

Autocorrelation values are four-valued or five-valued depending on parity conditions.

The sequence has quite good linear complexity.

Constructed cyclic codes have lower bounds on minimum distance.

Abstract

Binary Whiteman's cyclotomic sequences of orders 2 and 4 have a number of good randomness properties. In this paper, we compute the autocorrelation values and linear complexity of the first class two-prime Whiteman's generalized cyclotomic sequence (WGCS-I) of order $d=6$. Our results show that the autocorrelation values of this sequence is four-valued or five-valued if $(n_1-1)(n_2-1)/36$ is even or odd respectively, where $n_1$ and $n_2$ are two distinct odd primes and their linear complexity is quite good. We employ the two-prime WGCS-I of order 6 to construct several classes of cyclic codes over $\mathrm{GF}(q)$ with length $n_1n_2$. We also obtain the lower bounds on the minimum distance of these cyclic codes.