Optimal Binary LCD Codes

TL;DR

This paper investigates properties and bounds of binary LCD codes, providing new inequalities, classification results, and tables of maximum minimum weights for various code parameters.

Contribution

It introduces new properties of binary LCD codes via shortened and punctured codes, and provides bounds, classifications, and tables for their maximum minimum weights.

Findings

Derived inequalities for $d_{LCD}(n,k)$

Provided tables of $d_{LCD}(n,k)$ for $k extless=32$, $n extless=40

Classified binary LCD codes for certain parameters

Abstract

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1964. Nowadays, LCD codes are extensively studied in the literature and widely applied in data storage, cryptography, etc. In this paper, we prove some properties of binary LCD codes using their shortened and punctured codes. We also present some inequalities for the largest minimum weight $d_{LCD}(n,k)$ of binary LCD [n,k] codes for given length n and dimension k. Furthermore, we give two tables with the values of $d_{LCD}(n,k)$ for $k\le 32$ and $n\le 40$, and two tables with classification results.