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Linear codes in the folded Hamming distance and the quasi MDS property

Umberto Martínez-Peñas, Rubén Rodríguez-Ballesteros

Abstract

In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field sizes, which beats any MDS code.

Linear codes in the folded Hamming distance and the quasi MDS property

Abstract

In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field sizes, which beats any MDS code.
Paper Structure (7 sections, 34 theorems, 51 equations)

This paper contains 7 sections, 34 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The folded Hamming distance
  3. Quasi MDS codes
  4. Existence and density of dually QMDS codes
  5. Weight distributions
  6. Long QMDS codes over small fields
  7. Pseudo arcs: A geometric description

Key Result

Proposition 3

Let $\mathcal{C} \subseteq \mathbb{F}_{q^r}^n$ be an $\mathbb{F}_{q^r}$-linear code and let $\mathcal{C}^\perp \subseteq \mathbb{F}_{q^r}^n$ denote its dual with respect to the usual inner product in $\mathbb{F}_{q^r}^n$. If $\boldsymbol\beta = (\beta_1, \ldots, \beta_r)$ and $\boldsymbol\alpha = (\ (Dual bases of $\mathbb{F}_{q^r}$ over $\mathbb{F}_q$ always exist, see lidl.)

Theorems & Definitions (83)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Proposition 4: sr-hamming
  • Definition 5
  • Corollary 6
  • Proposition 7
  • Definition 8
  • Proposition 9: ball-additiveblaum-lowestxu
  • ...and 73 more