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Griesmer Bound and Constructions of Linear Codes in $b$-Symbol Metric

Gaojun Luo, Martianus Frederic Ezerman, Cem Güneri, San Ling, Ferruh Özbudak

TL;DR

This paper presents the b-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes and proposes two families of distance-optimal linear codes with respect to the b-symbol Griesmer bound.

Abstract

The $b$-symbol metric is a generalization of the Hamming metric. Linear codes, in the $b$-symbol metric, have been used in the read channel whose outputs consist of $b$ consecutive symbols. The Griesmer bound outperforms the Singleton bound for $\mathbb{F}_q$-linear codes in the Hamming metric, when $q$ is fixed and the length is large enough. This scenario is also applicable in the $b$-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the $b$-symbol metric. In this paper, we present the $b$-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the $b$-symbol Griesmer bound.

Griesmer Bound and Constructions of Linear Codes in $b$-Symbol Metric

TL;DR

This paper presents the b-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes and proposes two families of distance-optimal linear codes with respect to the b-symbol Griesmer bound.

Abstract

The -symbol metric is a generalization of the Hamming metric. Linear codes, in the -symbol metric, have been used in the read channel whose outputs consist of consecutive symbols. The Griesmer bound outperforms the Singleton bound for -linear codes in the Hamming metric, when is fixed and the length is large enough. This scenario is also applicable in the -symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the -symbol metric. In this paper, we present the -symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the -symbol Griesmer bound.
Paper Structure (8 sections, 7 theorems, 46 equations)

This paper contains 8 sections, 7 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Linear Codes in the $b$-Symbol Metric
  3. Known Results
  4. Our Contributions
  5. Preliminaries
  6. Griesmer Bound for Codes in the $b$-Symbol Metric
  7. Constructions of Linear Codes in $b$-Symbol Metric
  8. Concluding remarks

Key Result

Theorem 1

If ${\mathcal{C}}$ is any $(n,k,d_b)^b_q$ linear code, then

Theorems & Definitions (11)

  • Conjecture 1
  • Theorem 1: $b$-symbol Griesmer Bound
  • Theorem 2
  • Example 1
  • Remark 1
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Remark 2
  • Theorem 6
  • ...and 1 more