LCD Codes from tridiagonal Toeplitz matrice

TL;DR

This paper investigates LCD codes derived from double Toeplitz matrices, specifically focusing on tridiagonal symmetric cases, and introduces a construction method for optimal binary and ternary LCD codes.

Contribution

It explicitly determines the spectrum of tridiagonal symmetric Toeplitz matrices using Dickson polynomials and develops a concatenation process to construct optimal LCD codes.

Findings

Explicit spectrum formula for tridiagonal symmetric Toeplitz matrices

Conditions for codes to be LCD based on spectrum analysis

Construction of optimal or quasi-optimal binary and ternary LCD codes

Abstract

Double Toeplitz (DT) codes are codes with a generator matrix of the form $(I,T)$ with $T$ a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When $T$ is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.