Two classes of reducible cyclic codes with large minimum symbol-pair distances

TL;DR

This paper investigates two classes of reducible cyclic codes under the symbol-pair metric, revealing their weight distributions, minimum symbol-pair distances, and constructing MDS symbol-pair codes using cyclotomic theory.

Contribution

It introduces new classes of reducible cyclic codes with large minimum symbol-pair distances and constructs MDS symbol-pair codes using cyclotomic numbers and Gaussian periods.

Findings

Minimum symbol-pair distances are twice the Hamming distances under certain conditions.

Identified possible symbol-pair weights for the codes.

Constructed new three symbol-pair weight codes and determined their distributions.

Abstract

The high-density data storage technology aims to design high-capacity storage at a relatively low cost. In order to achieve this goal, symbol-pair codes were proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output pairs of overlapping symbols. Such a channel is called symbol-pair read channel, which introduce new concept called symbol-pair weight and minimum symbol-pair distance. In this paper, we consider the parameters of two classes of reducible cyclic codes under the symbol-pair metric. Based on the theory of cyclotomic numbers and Gaussian period over finite fields, we show the possible symbol-pair weights of these codes. Their minimum symbol-pair distances are twice the minimum Hamming distances under some conditions. Moreover, we obtain some three symbol-pair weight codes and determine their symbol-pair weight distribution. A class of MDS symbol-pair codes is also established. Among other results, we determine the values of some generalized cyclotomic numbers.