Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two families of linear codes with desirable properties from some functions over finite fields

Ziling Heng, Xiaoru Li, Yansheng Wu, Qi Wang

TL;DR

The paper addresses constructing linear codes over ${\mathbb F}_q$ with favorable locality, extendability, and self-orthogonality properties by leveraging augmentation techniques. It introduces two unbounded-length code families: the first from monomial-defining sets $D=\{x\in{\mathbb F}_{q^m}: {\mathrm{Tr}}_{q^t/q}(x^N)=0\}$ and the second from weakly regular bent functions, and proves locality results (typically ${\rm loc}=2$ for the first family with $q>2$, and ${\rm loc}=3$ in selected cases for the second family) along with self-orthogonality and extendability. The weight distributions, dual distances, and conditions for optimal or almost-optimal locally recoverable codes are derived, with several cases yielding 2-designs and numerous explicit parameter tables. These constructions provide scalable, high-rate LRCs suitable for distributed storage, cryptography, and related design-theoretic applications, and they establish a framework linking augmentation, bent-function theory, and combinatorial designs to code performance.

Abstract

Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic. In this paper, based on the augmentation technique, we present two families of linear codes from some functions over finite fields. The first family of linear codes is constructed from monomial functions over finite fields. The locality of them is determined and the weight distributions of two subfamilies of the codes are also given. An infinite family of locally recoverable codes which are at least almost optimal and some optimal recoverable codes are obtained from the linear codes. In particular, the two subfamilies of the codes are proved to be both optimally or almost optimally extendable and self-orthogonal. The second family of linear codes is constructed from weakly regular bent functions over finite fields and their weight distribution is determined. This family of codes is proved to have locality 3 for some cases and is conjectured to have locality 2 for other cases. Particularly, two families of optimal locally recoverable codes are derived from the linear codes. Besides, this family of codes is also proved to be both optimally or almost optimally extendable and self-orthogonal.

Two families of linear codes with desirable properties from some functions over finite fields

TL;DR

The paper addresses constructing linear codes over with favorable locality, extendability, and self-orthogonality properties by leveraging augmentation techniques. It introduces two unbounded-length code families: the first from monomial-defining sets and the second from weakly regular bent functions, and proves locality results (typically for the first family with , and in selected cases for the second family) along with self-orthogonality and extendability. The weight distributions, dual distances, and conditions for optimal or almost-optimal locally recoverable codes are derived, with several cases yielding 2-designs and numerous explicit parameter tables. These constructions provide scalable, high-rate LRCs suitable for distributed storage, cryptography, and related design-theoretic applications, and they establish a framework linking augmentation, bent-function theory, and combinatorial designs to code performance.

Abstract

Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic. In this paper, based on the augmentation technique, we present two families of linear codes from some functions over finite fields. The first family of linear codes is constructed from monomial functions over finite fields. The locality of them is determined and the weight distributions of two subfamilies of the codes are also given. An infinite family of locally recoverable codes which are at least almost optimal and some optimal recoverable codes are obtained from the linear codes. In particular, the two subfamilies of the codes are proved to be both optimally or almost optimally extendable and self-orthogonal. The second family of linear codes is constructed from weakly regular bent functions over finite fields and their weight distribution is determined. This family of codes is proved to have locality 3 for some cases and is conjectured to have locality 2 for other cases. Particularly, two families of optimal locally recoverable codes are derived from the linear codes. Besides, this family of codes is also proved to be both optimally or almost optimally extendable and self-orthogonal.
Paper Structure (19 sections, 37 theorems, 124 equations, 8 tables)

This paper contains 19 sections, 37 theorems, 124 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Linear codes and self-orthogonal codes
  3. Optimally extendable codes
  4. Locally recoverable codes
  5. The objectives of this paper
  6. Preliminaries
  7. Characters over finite fields
  8. Gaussian sums over finite fields
  9. The number of roots of quadratic polynomial over finite fields
  10. Weakly regular bent functions and the cyclotomic fields
  11. The first family of linear codes $\overline{{\mathcal{C}}_D}$
  12. The dual distance and the locality of $\overline{{\mathcal{C}}_D}$
  13. The case that $N=q^r+1$
  14. The case that $N=2$
  15. The second family of linear codes ${\mathcal{C}}_f$ from weakly regular bent functions
  16. ...and 4 more sections

Key Result

Lemma 1

H Let $M$ be the maximum number of codewords in a code over ${\mathbb{F}}_q$ of length $n$ and minimum distance $d$. Then where $t=\lfloor(d-1)/2\rfloor$.

Theorems & Definitions (79)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5: GH, Singleton-like bound
  • Lemma 6: Upper, Cadambe-Mazumdar bound
  • Lemma 7: L, Theorem 5.15
  • Lemma 8: L, Theorem 5.33
  • Lemma 9
  • Lemma 10
  • ...and 69 more