Cyclic Codes from Cyclotomic Sequences of Order Four

TL;DR

This paper constructs cyclic codes from cyclotomic sequences of order four, achieving many optimal or near-optimal codes with applications in data transmission and electronics.

Contribution

It introduces new classes of cyclic codes over finite fields using cyclotomic sequences of order four, with bounds on minimum weight and connections to difference sets.

Findings

Some codes are optimal or almost optimal.

Lower bounds on minimum weight are established.

Codes are related to difference sets and almost difference sets.

Abstract

Cyclic codes are an interesting subclass of linear codes and have been used in consumer electronics, data transmission technologies, broadcast systems, and computer applications due to their efficient encoding and decoding algorithms. In this paper, three cyclotomic sequences of order four are employed to construct a number of classes of cyclic codes over $\gf(q)$ with prime length. Under certain conditions lower bounds on the minimum weight are developed. Some of the codes obtained are optimal or almost optimal. In general, the cyclic codes constructed in this paper are very good. Some of the cyclic codes obtained in this paper are closely related to almost difference sets and difference sets. As a byproduct, the $p$-rank of these (almost) difference sets are computed.