Constructions of optimal LCD codes over large finite fields

TL;DR

This paper proves the existence of optimal LCD codes over large finite fields and introduces various methods for their construction, including random sampling, code extension, and matrix products.

Contribution

It provides new existence proofs and construction techniques for optimal LCD codes over large finite fields, expanding the toolkit for code design.

Findings

Existence of optimal LCD codes over large finite fields established.

Multiple construction methods demonstrated, including random sampling and matrix product codes.

Framework for generating orthogonal matrices over finite fields developed.

Abstract

In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.