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Minimum distances of binary optimal LCD codes of dimension five are completely determined

Yang Liu, Ruihu Li, Qiang Fu, Hao Song

TL;DR

Combined with known results on optimal LCD code, d_{l}(n,5)$ of all $[ n,5]$ LCD codes are completely determined.

Abstract

Let $t \in \{2,8,10,12,14,16,18\}$ and $n=31s+t\geq 14$, $d_{a}(n,5)$ and $d_{l}(n,5)$ be distances of binary $[n,5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n,5,d_{a}(n,5)]$ optimal linear code is not an LCD code, there is an $[n,5,d_{l}(n,5)]=[n,5,d_{a}(n,5)-1]$ optimal LCD code if $t\neq 16$, and an optimal $[n,5,d_{l}(n,5)]$ optimal LCD code has $d_{l}(n,5)=16s+6=d_{a}(n,5)-2$ for $t=16$. Combined with known results on optimal LCD code, $d_{l}(n,5)$ of all $[n,5]$ LCD codes are completely determined.

Minimum distances of binary optimal LCD codes of dimension five are completely determined

TL;DR

Combined with known results on optimal LCD code, d_{l}(n,5)[ n,5]$ LCD codes are completely determined.

Abstract

Let and , and be distances of binary optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an optimal linear code is not an LCD code, there is an optimal LCD code if , and an optimal optimal LCD code has for . Combined with known results on optimal LCD code, of all LCD codes are completely determined.
Paper Structure (8 sections, 15 theorems, 8 equations, 7 tables)

This paper contains 8 sections, 15 theorems, 8 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['the1.1']}
  4. $h([32s+2,5, d_{a}])\geq 1$ and $h([32s+t ,5, d_{a}])\geq 2$ for $t=8, 12, 16$
  5. $h([31s+10,5, 16s+4]) \geq 1$
  6. $h([31s+14,5, 16s+6])\geq 1$
  7. $h([31s+18,5, 16s+8])\geq 1$
  8. Conclusion

Key Result

Theorem 1

If $s$ is an integer, $t\in \{2, 8,10,12, 14,16,18\}$ and $n=31s+t\geq 14$, then an optimal $[n,5,d_{a}(n,5)]$ linear code is not an LCD code, and $d_{l}(n,5)=d_{a}-1$ if $t\neq 16$ and $d_{l}(n,5)=d_{a}-2$ if $t= 16$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Proposition 3
  • Lemma 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 16 more