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Self-Orthogonal Codes from Vectorial Dual-Bent Functions

Jiaxin Wang, Yadi Wei, Fang-Wei Fu, Juan Li

Abstract

Self-orthogonal codes are a significant class of linear codes in coding theory and have attracted a lot of attention. In \cite{HLL2023Te,LH2023Se}, $p$-ary self-orthogonal codes were constructed by using $p$-ary weakly regular bent functions, where $p$ is an odd prime. In \cite{WH2023Se}, two classes of non-degenerate quadratic forms were used to construct $q$-ary self-orthogonal codes, where $q$ is a power of a prime. In this paper, we construct new families of $q$-ary self-orthogonal codes using vectorial dual-bent functions. Some classes of at least almost optimal linear codes are obtained from the dual codes of the constructed self-orthogonal codes. In some cases, we completely determine the weight distributions of the constructed self-orthogonal codes. From the view of vectorial dual-bent functions, we illustrate that the works on constructing self-orthogonal codes from $p$-ary weakly regular bent functions \cite{HLL2023Te,LH2023Se} and non-degenerate quadratic forms with $q$ being odd \cite{WH2023Se} can be obtained by our results. We partially answer an open problem on determining the weight distribution of a class of self-orthogonal codes given in \cite{LH2023Se}. As applications, we construct new infinite families of at least almost optimal $q$-ary linear complementary dual codes (for short, LCD codes) and quantum codes.

Self-Orthogonal Codes from Vectorial Dual-Bent Functions

Abstract

Self-orthogonal codes are a significant class of linear codes in coding theory and have attracted a lot of attention. In \cite{HLL2023Te,LH2023Se}, -ary self-orthogonal codes were constructed by using -ary weakly regular bent functions, where is an odd prime. In \cite{WH2023Se}, two classes of non-degenerate quadratic forms were used to construct -ary self-orthogonal codes, where is a power of a prime. In this paper, we construct new families of -ary self-orthogonal codes using vectorial dual-bent functions. Some classes of at least almost optimal linear codes are obtained from the dual codes of the constructed self-orthogonal codes. In some cases, we completely determine the weight distributions of the constructed self-orthogonal codes. From the view of vectorial dual-bent functions, we illustrate that the works on constructing self-orthogonal codes from -ary weakly regular bent functions \cite{HLL2023Te,LH2023Se} and non-degenerate quadratic forms with being odd \cite{WH2023Se} can be obtained by our results. We partially answer an open problem on determining the weight distribution of a class of self-orthogonal codes given in \cite{LH2023Se}. As applications, we construct new infinite families of at least almost optimal -ary linear complementary dual codes (for short, LCD codes) and quantum codes.
Paper Structure (24 sections, 32 theorems, 99 equations, 17 tables)

This paper contains 24 sections, 32 theorems, 99 equations, 17 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Some results on vectorial dual-bent functions
  5. Some results on linear codes
  6. Some results on character sums
  7. Self-orthogonal codes from vectorial dual-bent functions with Condition I
  8. Some lemmas
  9. Self-orthogonal codes constructed from vectorial dual-bent functions with Condition I
  10. Self-orthogonal codes from vectorial dual-bent functions with Condition II
  11. Some lemmas
  12. Self-orthogonal codes constructed from vectorial dual-bent functions with Condition II
  13. Self-orthogonal codes from vectorial dual-bent functions with Condition III
  14. Some lemmas
  15. Self-orthogonal codes constructed from vectorial dual-bent functions with Condition III
  16. ...and 9 more sections

Key Result

Proposition 1

Let $f: V_{n}^{(p)}\rightarrow \mathbb{F}_{p}$ be a bent function belonging to $\mathcal{RF}$, that is, $f$ is a weakly regular bent function of $l$-form with $f(0)=0$ and $gcd(l-1, p-1)=1$ for some $l$. Then $f$ (seen as a vectorial bent function from $V_{n}^{(p)}$ to $V_{1}^{(p)}$) is a vectorial

Theorems & Definitions (61)

  • Proposition 1: CM2018Be
  • Proposition 2: Hamming Bound
  • Proposition 3: Wan1998A
  • Proposition 4: Massey1998Or
  • Proposition 5: Quantum Hamming Bound KKK2006No
  • Proposition 6: Steane construction Hamada2008Co
  • Proposition 7: LN1997Fi
  • Proposition 8: LN1997Fi
  • Proposition 9: Ding2015Co
  • Proposition 10: Ding2015Co
  • ...and 51 more