Monimial Matrix Analogue of Yoshida's theorem
Ananda Chakraborty
TL;DR
The paper addresses extending weight-enumerator dualities for $q$-ary linear codes to average complete joint weight enumerators and their monomial-equivalence classes. It develops MacWilliams-type identities for the averaged joint weight enumerators and proves a monomial-matrix analogue of Yoshida's theorem, first for two codes and then for the average of $g$-fold joint weight enumerators. The main contributions include explicit averaged formulas that link dual and primal enumerators via monomial transformations and composition structures, and a broad generalization to $g$-fold configurations. These results deepen the understanding of duality, monomial equivalence, and weight enumerator behavior in finite-field linear codes, with potential implications for code classification and combinatorial enumeration in coding theory.
Abstract
In this paper, we study variants of weight enumerators of linear codes over $\mathbb{F}_q$. We generalize the concept of average complete joint weight enumerators of two linear codes over $\mathbb{F}_q$. We also give its MacWilliams type identities. Then we establish a monomial analogue of Yoshida's theorem for this average complete joint weight enumerators. Finally, we present the generalized representation for average of $g$-fold complete joint weight enumerators for $\mathbb{F}_q$-linear codes and establish a monomial matrix analogue of Yoshida's theorem for average $g$-fold complete joint weight enumerators.