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A family of self-orthogonal divisible codes with locality 2

Ziling Heng, Mengjie Yang, Yang Ming

TL;DR

The paper addresses constructing self-orthogonal divisible linear codes with locality 2 using a trace and norm based defining set $D$ and its augmentation $\overline{\mathcal{C}_{D}}$. It analyzes weight distributions via Gaussian sums and exponential sums $\Delta(b)$ and $\Omega(b,c)$, deriving three-, four-, and five-weight codes under various parameter regimes. A key result is that $\overline{\mathcal{C}_{D}}$ is $q$-divisible and self-orthogonal for suitable $m_{1},m_{2}$ and odd $q$, and locality 2 holds for $q>2$, yielding numerous optimal or near-optimal locally recoverable codes and distance-optimal binary examples. These codes have practical impact for distributed storage and lattice-based constructions, and the methodology connects trace–norm algebra with weight distributions through Gaussian sums.

Abstract

Linear codes are widely studied due to their applications in communication, cryptography, quantum codes, distributed storage and many other fields. In this paper, we use the trace and norm functions over finite fields to construct a family of linear codes. The weight distributions of the codes are determined in three cases via Gaussian sums. The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases. In particular, we prove that this family of linear codes has locality 2. Several optimal or almost optimal linear codes and locally recoverable codes are derived. In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained. The self-orthogonal codes derived in this paper can be used to construct lattices and have nice application in distributed storage.

A family of self-orthogonal divisible codes with locality 2

TL;DR

The paper addresses constructing self-orthogonal divisible linear codes with locality 2 using a trace and norm based defining set and its augmentation . It analyzes weight distributions via Gaussian sums and exponential sums and , deriving three-, four-, and five-weight codes under various parameter regimes. A key result is that is -divisible and self-orthogonal for suitable and odd , and locality 2 holds for , yielding numerous optimal or near-optimal locally recoverable codes and distance-optimal binary examples. These codes have practical impact for distributed storage and lattice-based constructions, and the methodology connects trace–norm algebra with weight distributions through Gaussian sums.

Abstract

Linear codes are widely studied due to their applications in communication, cryptography, quantum codes, distributed storage and many other fields. In this paper, we use the trace and norm functions over finite fields to construct a family of linear codes. The weight distributions of the codes are determined in three cases via Gaussian sums. The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases. In particular, we prove that this family of linear codes has locality 2. Several optimal or almost optimal linear codes and locally recoverable codes are derived. In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained. The self-orthogonal codes derived in this paper can be used to construct lattices and have nice application in distributed storage.
Paper Structure (9 sections, 12 theorems, 91 equations, 3 tables)

This paper contains 9 sections, 12 theorems, 91 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Characters of finite fields and Gaussian sums
  5. Some exponential sums
  6. The weight distribution and the locality of $\overline{\mathcal{C}_{D}}$
  7. The weight distribution of $\overline{\mathcal{C}_{D}}$
  8. The locality of $\overline{\mathcal{C}_{D}}$
  9. Concluding remarks

Key Result

Lemma 1

LH Let $q=p^s$, 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 (29)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 19 more