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The augmented codes of a family of linear codes with locality 2

Ziling Heng, Keqing Cao

TL;DR

This work generalizes a family of linear codes introduced by Ding and Ding and introduces augmented versions to improve structure and utility. By employing trace-based defining sets and Gaussian-sum techniques, the authors derive explicit parameter sets and weight distributions for the augmented codes in two divisibility scenarios, proving $p$-divisibility, self-orthogonality, locality $2$, and projectivity. The results yield several (almost) optimal linear codes and locally recoverable codes, with improved rates over prior constructions and concrete examples validating optimality in some cases. The findings suggest strong applicability to distributed storage and potential relevance to quantum codes and lattices, thanks to self-orthogonality and projectivity. Overall, the paper advances code constructions with favorable weight structures, locality, and dual-distance properties that support practical encoding/decoding in storage and communication systems.

Abstract

In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums to determine the parameters and weight distributions of the augmented codes in some cases. It is shown that the augmented codes are self-orthogonal and have only a few nonzero weights. For another thing, the locality of the augmented codes is proved to be 2, which indicates the augmented codes are useful in distributed storage. Besides, the augmented codes are projective as the minimum distance of their duals is proved to be 3. In particular, we obtain several (almost) optimal linear codes and locally recoverable codes.

The augmented codes of a family of linear codes with locality 2

TL;DR

This work generalizes a family of linear codes introduced by Ding and Ding and introduces augmented versions to improve structure and utility. By employing trace-based defining sets and Gaussian-sum techniques, the authors derive explicit parameter sets and weight distributions for the augmented codes in two divisibility scenarios, proving -divisibility, self-orthogonality, locality , and projectivity. The results yield several (almost) optimal linear codes and locally recoverable codes, with improved rates over prior constructions and concrete examples validating optimality in some cases. The findings suggest strong applicability to distributed storage and potential relevance to quantum codes and lattices, thanks to self-orthogonality and projectivity. Overall, the paper advances code constructions with favorable weight structures, locality, and dual-distance properties that support practical encoding/decoding in storage and communication systems.

Abstract

In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums to determine the parameters and weight distributions of the augmented codes in some cases. It is shown that the augmented codes are self-orthogonal and have only a few nonzero weights. For another thing, the locality of the augmented codes is proved to be 2, which indicates the augmented codes are useful in distributed storage. Besides, the augmented codes are projective as the minimum distance of their duals is proved to be 3. In particular, we obtain several (almost) optimal linear codes and locally recoverable codes.
Paper Structure (13 sections, 14 theorems, 70 equations, 7 tables)

This paper contains 13 sections, 14 theorems, 70 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Linear codes
  3. Locally recoverable codes
  4. The objectives of this paper
  5. Preliminaries
  6. Characters over finite fields
  7. Gaussian sums
  8. Some lemmas
  9. The weight distribution and self-orthogonality of $\overline{{\mathcal{C}}_D}$
  10. The case that $m_2 \mid m_1\mid m$
  11. The case that $m_1 \mid m_2\mid m$
  12. The locality of $\overline{{\mathcal{C}}_D}$ and the minimum distance of $\overline{{\mathcal{C}}_D}^\perp$
  13. Concluding remarks

Key Result

Lemma 1

LH Let $q=p^e$, where $p$ is an odd prime. Let ${\mathcal{C}}$ be an $[n,k,d]$ linear code over ${\mathbb{F}}_q$ with $\mathbf{1} \in{\mathcal{C}}$, where $\mathbf{1}$ is all-1 vector of length $n$. If the linear code ${\mathcal{C}}$ is $p$-divisible, then ${\mathcal{C}}$ is self-orthogonal.

Theorems & Definitions (30)

  • Lemma 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4: L, Theorem 5.15
  • Lemma 5: L, Theorem 5.33
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 20 more