Characterization and classification of optimal LCD codes

TL;DR

This paper characterizes and classifies optimal linear complementary dual (LCD) codes over finite fields, determining their maximum minimum weights and providing complete classifications for specific parameters.

Contribution

It offers a new characterization of LCD codes with large minimum weights and classifies optimal LCD codes for certain lengths and dimensions over finite fields.

Findings

Largest minimum weights identified for specific LCD codes

Complete classification of optimal LCD codes for given parameters

Characterization of LCD codes with large minimum weights

Abstract

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over $\mathbb{F}_q$ having large minimum weights for $q \in \{2,3\}$. Using the characterization, we determine the largest minimum weights among LCD $[n,k]$ codes over $\mathbb{F}_q$ for $(q,k) \in \{(2,4), (3,2),(3,3)\}$. Moreover, we give a complete classification of optimal LCD $[n,k]$ codes over $\mathbb{F}_q$ for $(q,k) \in \{(2,3), (2,4), (3,2),(3,3)\}$.