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Self-orthogonal codes from plateaued functions

Peng Wang, Ziling Heng

TL;DR

This family of linear codes with four nonzero weights is proved to be not only self-orthogonal but also optimally or almost optimally extendable and derive binary and ternary linearly complementary dual codes with new parameters from this family of codes.

Abstract

Self-orthogonal codes are of interest as they have important applications in quantum codes, lattices and many areas. In this paper, based on the weakly regular plateaued functions or plateaued Boolean functions, we construct a family of linear codes with four nonzero weights. This family of linear codes is proved to be not only self-orthogonal but also optimally or almost optimally extendable. Besides, we derive binary and ternary linearly complementary dual codes (LCD codes for short) with new parameters from this family of codes. Some families of self-dual codes are also obtained as byproducts.

Self-orthogonal codes from plateaued functions

TL;DR

This family of linear codes with four nonzero weights is proved to be not only self-orthogonal but also optimally or almost optimally extendable and derive binary and ternary linearly complementary dual codes with new parameters from this family of codes.

Abstract

Self-orthogonal codes are of interest as they have important applications in quantum codes, lattices and many areas. In this paper, based on the weakly regular plateaued functions or plateaued Boolean functions, we construct a family of linear codes with four nonzero weights. This family of linear codes is proved to be not only self-orthogonal but also optimally or almost optimally extendable. Besides, we derive binary and ternary linearly complementary dual codes (LCD codes for short) with new parameters from this family of codes. Some families of self-dual codes are also obtained as byproducts.

Paper Structure

This paper contains 15 sections, 27 theorems, 49 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Linear codes and self-orthogonal codes
  3. LCD codes
  4. Optimally and almost optimally extendable codes
  5. The purpose of this paper
  6. Preliminaries
  7. Characters and Gaussian sums over finite fields
  8. Weakly regular plateaued functions and the cyclotomic fields
  9. Cyclotomic field
  10. Four-weight linear codes from $p$-ary weakly regular $s$-plateaued functions
  11. The weight distribution of $\overline{{\mathcal{C}}_f}$
  12. The optimally extendability of $\overline{{\mathcal{C}}_f}$
  13. Binary self-orthogonal codes and LCD codes from $s$-plateaued Boolean functions
  14. Self-dual codes
  15. Summary and concluding remarks

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 $t=\lfloor(d-1)/2\rfloor$ and $\lfloor \cdot \rfloor$ denotes the floor function.

Theorems & Definitions (42)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 32 more