The joint weight enumerator of an LCD code and its dual

TL;DR

This paper introduces a linear programming bound for LCD codes based on the joint weight enumerator with their duals, revealing algebraic invariants and symmetries that could inform code design and analysis.

Contribution

It establishes linear constraints on the joint weight enumerator of LCD codes and their duals, and explores its invariance properties and potential Gleason-like formulas.

Findings

Derived linear constraints lead to new bounds on LCD code size.

Identified invariance of the joint weight enumerator under a specific matrix group.

Proposed a sketch of a Gleason formula for the joint weight enumerator.

Abstract

A binary linear code is called {\em LCD} if it intersects its dual trivially. We show that the coefficients of the joint weight enumerator of such a code with its dual satisfy linear constraints, leading to a new linear programming bound on the size of an LCD code of given length and minimum distance. In addition, we show that this polynomial is, in general, an invariant of a matrix group of dimension $4$ and order $12$. Also, we sketch a Gleason formula for this weight enumerator.