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Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes

Hao Chen, Yaqi Chen, Conghui Xie, Huimin Lao

TL;DR

The paper addresses constructing minimal $q$-ary linear codes that violate the Ashikhmin-Barg condition. It develops a general augmentation method to turn AB-satisfying minimal codes into AB-violating ones and shows how to obtain AB-violating minimal codes from arbitrary projective codes using simplex-complementary constructions; it then furnishes infinitely many explicit families (including self-orthogonal binary ones) and determines their weight distributions. The key contributions are Theorem ${\bf 2.1}$ (AB-violating augmentation), Theorem ${\bf 3.1}$ (simplex-complementary AB-satisfying construction from projective codes), and numerous infinite families with near-optimal parameters, along with explicit weight-distribution formulas. These results expand the toolkit for generating AB-violating minimal codes and highlight rich connections between projective codes, simplex complements, and minimality with practical implications for coding theory and related applications.

Abstract

In recent years, there have been many constructions of minimal linear codes violating the Ashikhmin-Barg condition from Boolean functions, linear codes with few nonzero weights or partial difference sets. In this paper, we first give a general method to transform a minimal code satisfying the Ashikhmin-Barg condition to a minimal code violating the Ashikhmin-Barg condition. Then we give a construction of a minimal code satisfying the Ashikhmin-Barg condition from an arbitrary projective linear code. Hence an arbitrary projective linear code can be transformed to a minimal codes violating the Ashikhmin-Barg condition. Then we give infinite many families of minimal codes violating the Ashikhamin-Barg condition. Weight distributions of constructed minimal codes violating the Ashikhmin-Barg condition in this paper are determined. Many minimal linear codes violating the Ashikhmin-Barg condition with their minimum weights close to the optimal or the best known minimum weights of linear codes are constructed in this paper. Moreover, many infinite families of self-orthogonal binary minimal codes violating the Ashikhmin-Barg condition are also given.

Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes

TL;DR

The paper addresses constructing minimal -ary linear codes that violate the Ashikhmin-Barg condition. It develops a general augmentation method to turn AB-satisfying minimal codes into AB-violating ones and shows how to obtain AB-violating minimal codes from arbitrary projective codes using simplex-complementary constructions; it then furnishes infinitely many explicit families (including self-orthogonal binary ones) and determines their weight distributions. The key contributions are Theorem (AB-violating augmentation), Theorem (simplex-complementary AB-satisfying construction from projective codes), and numerous infinite families with near-optimal parameters, along with explicit weight-distribution formulas. These results expand the toolkit for generating AB-violating minimal codes and highlight rich connections between projective codes, simplex complements, and minimality with practical implications for coding theory and related applications.

Abstract

In recent years, there have been many constructions of minimal linear codes violating the Ashikhmin-Barg condition from Boolean functions, linear codes with few nonzero weights or partial difference sets. In this paper, we first give a general method to transform a minimal code satisfying the Ashikhmin-Barg condition to a minimal code violating the Ashikhmin-Barg condition. Then we give a construction of a minimal code satisfying the Ashikhmin-Barg condition from an arbitrary projective linear code. Hence an arbitrary projective linear code can be transformed to a minimal codes violating the Ashikhmin-Barg condition. Then we give infinite many families of minimal codes violating the Ashikhamin-Barg condition. Weight distributions of constructed minimal codes violating the Ashikhmin-Barg condition in this paper are determined. Many minimal linear codes violating the Ashikhmin-Barg condition with their minimum weights close to the optimal or the best known minimum weights of linear codes are constructed in this paper. Moreover, many infinite families of self-orthogonal binary minimal codes violating the Ashikhmin-Barg condition are also given.
Paper Structure (15 sections, 21 theorems, 36 equations)

This paper contains 15 sections, 21 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related works
  4. Our contributions
  5. Minimal codes violating the Ashikhmin-Barg condition from codes satisfying the Ashikhmin-Barg condition
  6. Minimal codes satisfying the Ashikhmin-Barg condition from arbitrary projective linear codes
  7. Infinitely many families of minimal linear codes violating the Ashikhmin-Barg codndition
  8. From minimal Solomon-Stiffler codes
  9. From few-weight codes satisfying the Ashikhmin-Barg condition
  10. From linear codes violating the Ashikhmin-Barg condition
  11. From arbitrary projective linear codes
  12. Infinitely many families of self-orthogonal binary minimal linear codes violating the Ashikhmin-Barg condition
  13. Weight distributions of some minimal codes violating the Ashikhmin-Barg condition
  14. Conclusions
  15. Minimal codes violating Ashikhmin-Barg condition constructed in this paper

Key Result

Theorem 3.1

Let ${\bf C} \subset {\bf F}_q^n$ be a projective linear $[n,k]_q$ code with the maximum weight $w\leq n$. Let $h$ be a positive integer. Then the simplx complementary code of ${\bf C}$ has parameters $[\frac{q^{k+h}-1}{q-1}-n, k+h, q^{k+h-1}-w]_q$ and maximum weight $q^{k+h-1}$. When $h >\log_q w-k

Theorems & Definitions (37)

  • Theorem 3.1
  • Corollary 3.1
  • Proposition 4.1
  • Example 4.1
  • Example 4.2
  • Proposition 4.2
  • Example 4.3
  • Proposition 4.3
  • Example 4.4
  • Proposition 4.4
  • ...and 27 more