The connections among Hamming metric, $b$-symbol metric, and $r$-th generalized Hamming metric

TL;DR

This paper explores the relationships among Hamming, $b$-symbol, and $r$-th generalized Hamming metrics, proposing a conjecture on the $b$-symbol Griesmer Bound for cyclic codes and analyzing the $b$-symbol weight set.

Contribution

It establishes connections among three generalized Hamming metrics and introduces a conjecture on the $b$-symbol Griesmer Bound for cyclic codes.

Findings

Proposed a conjecture on the $b$-symbol Griesmer Bound.

Analyzed the combinatorial function of the $b$-symbol weight set.

Explored the relationships among the three metrics.

Abstract

The $r$-th generalized Hamming metric and the $b$-symbol metric are two different generalizations of Hamming metric. The former is used on the wire-tap channel of Type II, and the latter is motivated by the limitations of the reading process in high-density data storage systems and applied to a read channel that outputs overlapping symbols. In this paper, we study the connections among the three metrics (that is, Hamming metric, $b$-symbol metric, and $r$-th generalized Hamming metric) mentioned above and give a conjecture about the $b$-symbol Griesmer Bound for cyclic codes. %Furthermore, we explore the combinatorial function of the size of the $b$-symbol weight set of a code $C$.