Average of complete joint weight enumerators and self-dual codes

TL;DR

This paper provides a new representation for the average of complete joint weight enumerators of two linear codes over finite fields and rings, generalizing to multiple codes and analyzing intersection properties.

Contribution

It introduces a novel representation of average joint weight enumerators for pairs and groups of codes over different algebraic structures, extending previous results.

Findings

Derived a general formula for the average of complete joint weight enumerators

Extended the representation to g-fold joint weight enumerators

Calculated intersection numbers and their second moments for specific code types

Abstract

In this paper, we give a representation of the average of complete joint weight enumerators of two linear codes of length $n$ over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$ in terms of the compositions of $n$ and their distributions in the codes. We also obtain a generalization of the representation for the average of $g$-fold complete joint weight enumerators of codes over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$. Finally, the average of intersection numbers of a pair of Type III (resp. Type IV) codes, and its second moment are found.