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Optimal linear codes with few weights from simplicial complexes

Bing Chen, Yunge Xu, Zhao Hu, Nian Li, Xiangyong Zeng

Abstract

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let $q$ be a prime power. In this paper, by using the simplicial complexes of ${\mathbb F}_{q}^m$ with one single maximal element, we construct four families of linear codes over the ring ${\mathbb F}_{q}+u{\mathbb F}_{q}$ ($u^2=0$), which generalizes the results of [IEEE Trans. Inf. Theory 66(6):3657-3663, 2020]. The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over ${\mathbb F}_{q}$, including (near) Griesmer codes and distance-optimal codes.

Optimal linear codes with few weights from simplicial complexes

Abstract

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let be a prime power. In this paper, by using the simplicial complexes of with one single maximal element, we construct four families of linear codes over the ring (), which generalizes the results of [IEEE Trans. Inf. Theory 66(6):3657-3663, 2020]. The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over , including (near) Griesmer codes and distance-optimal codes.
Paper Structure (13 sections, 11 theorems, 76 equations)

This paper contains 13 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The definition of simplicial complexes of ${\mathbb F}_{q}^m$
  4. Linear codes over the ring $R$
  5. Useful auxiliary results
  6. Four families of linear codes over ${\mathbb F}_{q}+u{\mathbb F}_{q}$
  7. The first class of linear codes ${\mathcal{C}}_{L}$ with $L=\Delta_{A}+u(\Delta_{B}\setminus \Delta_{B'})$
  8. The second class of linear codes ${\mathcal{C}}_{L}$ with $L=\Delta_{A}^c+u(\Delta_{B}\setminus \Delta_{B'})$
  9. The third class of linear codes ${\mathcal{C}}_{L}$ with $L=\Delta_{A}+u(\Delta_{B}\setminus \Delta_{B'})^c$
  10. The fourth class of linear codes ${\mathcal{C}}_{L}$ with $L=\Delta_{A}^c+u(\Delta_{B}\setminus \Delta_{B'})^c$
  11. Optimal codes over ${\mathbb F}_{q}$ and examples
  12. Conclusions
  13. Acknowledgements

Key Result

Lemma 1

(HCLZ) Let $L=L_{1}+uL_{2}$ where $L_{1}, L_{2}\in {\mathbb F}_{q^m}$. Then ${\mathcal{C}}_{L}$ is a code of length $|L|$ over $R$ and for any $a+ub \in \mathcal{R}\backslash \{0\}$, the Lee weight of the codeword $c_{a+ub}$ in ${\mathcal{C}}_{L}$ is $wt_{L}(c_{a+ub})=2|L|-\Omega$ where

Theorems & Definitions (40)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Example 1
  • Theorem 2
  • proof
  • Remark 3
  • ...and 30 more