Cyclotomic Constructions of Cyclic Codes with Length Being the Product of Two Primes

TL;DR

This paper introduces new cyclic code constructions based on cyclotomy for lengths that are the product of two primes, providing bounds on their minimum weights and achieving optimal codes in some cases.

Contribution

It presents three types of generalized cyclotomy of order two and three classes of cyclic codes of length n1n2 with specific dimensions, advancing the design of efficient cyclic codes.

Findings

Constructed cyclic codes with length n1n2 and specific dimensions.

Established bounds on the minimum odd-like weight of these codes.

Produced some of the best cyclic codes in certain cases.

Abstract

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. They have been studied for decades and a lot of progress has been made. In this paper, three types of generalized cyclotomy of order two and three classes of cyclic codes of length $n_1n_2$ and dimension $(n_1n_2+1)/2$ are presented and analysed, where $n_1$ and $n_2$ are two distinct primes. Bounds on their minimum odd-like weight are also proved. The three constructions produce the best cyclic codes in certain cases.