The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices

TL;DR

This paper explores the combinatorial properties of LCD codes, providing new bounds and constructions using orthogonal matrices, self-dual codes, combinatorial designs, and Gray maps, advancing the understanding of their size and structure.

Contribution

It introduces a linear programming bound for the maximum size of LCD codes and constructs new LCD codes using various algebraic and combinatorial methods.

Findings

Linear programming bounds for LCD codes established

Construction methods for LCD codes using orthogonal matrices and designs

Lower bounds for code sizes provided for specific parameters

Abstract

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.