Double Toeplitz codes and their average weight enumerators

TL;DR

This paper investigates the average weight enumerators of double Toeplitz codes, a generalization of double circulant codes, and explores their existence and classification over finite fields for specific parameters.

Contribution

It introduces the study of average weight enumerators for double Toeplitz codes and provides existence results and classifications for codes with large minimum weights over finite fields.

Findings

Derived average weight enumerators for double Toeplitz codes.

Established existence of codes with specified minimum weights over  fields.

Classified codes with largest minimum weights for modest lengths.

Abstract

Recently, double Toeplitz codes have been introduced as a generalization of double circulant codes. In this paper, we study the average weight enumerators of double Toeplitz codes. As an application, we consider the existence of double Toeplitz codes over $\mathbb{F}_q$ with some specified minimum weights for $q \in \{2,3,4\}$. We also give a classification of double Toeplitz codes over $\mathbb{F}_q$ with the largest minimum weights for modest lengths and $q \in \{2,3,4\}$.