The b-symbol weight distribution of irreducible cyclic codes and related consequences

TL;DR

This paper derives a formula for the $b$-symbol weight distribution of irreducible cyclic codes, explores their weight hierarchies, and identifies optimal shortened codes, advancing coding theory for high-density data storage.

Contribution

It introduces a general $b$-symbol weight enumerator formula for irreducible cyclic codes using Gaussian periods and a new invariant, and analyzes their weight hierarchies and optimal shortened codes.

Findings

Derived the $b$-symbol weight enumerator formula for irreducible cyclic codes.

Determined the $b$-symbol weight hierarchies for certain cases.

Identified optimal shortened codes from irreducible cyclic codes.

Abstract

The $b$-symbol read channel is motivated by the limitations of the reading process in high density data storage systems. The corresponding new metric is a generalization of the Hamming metric known as the $b$-symbol weight metric and has become an important object in coding theory. In this paper, the general $b$-symbol weight enumerator formula for irreducible cyclic codes is presented by using the Gaussian period and a new invariant $\#U(b,j,N_1)$. The related $b$-symbol weight hierarchies $\{d_1(\C),d_2(\C),\ldots,d_K(\C)\}$ ($K=\dim(\C)$) are given for some cases. The shortened codes which are optimal from some classes of irreducible cyclic codes are given, where the shorten set $\mathcal{T}$ is the complementary set of $b$-symbol support of some codeword with the minimal $b$-symbol weight.