Gustavo T. Bastos | Searcharxiv

arXiv2026

Construction of Generalized Weighing-Hadamard Matrices over Finite Fields

The existence, several properties, and constructions of Generalized Weighing-Hadamard (GWH) matrices over finite fields are addressed in this work. We study the subset of invertible GWH matrices and show that it forms a group under matrix multiplication. Besides that, we introduce a strong notion of equivalence between such matrices, defined via orthogonal transformations, and further prove that the corresponding quotient group by the subgroup of orthogonal matrices is abelian. Finally, we discuss some applications of these matrices in coding theory

cs.IT↗

arXiv2025

Linearity of $\mathbb{Z}_{2^L}$-Linear Codes via Schur Product

We propose an innovative approach to investigating the linearity of $\mathbb{Z}_{2^L}$-linear codes derived from $\mathbb{Z}_{2^L}$-additive codes using the generalized Gray map. To achieve this, we define two related binary codes: the associated and the decomposition codes. By considering the Schur product between codewords, we can determine the linearity of the respective $\mathbb{Z}_{2^L}$-linear code. As a result, we establish a connection between the linearity of the $\mathbb{Z}_{2^L}$-linear codes with the linearity of the decomposition code for $\mathbb{Z}_4$ and $\mathbb{Z}_8$-additive codes. Furthermore, we construct $\mathbb{Z}_{2^L}$-additive codes from nested binary codes, resulting in linear $\mathbb{Z}_{2^L}$-linear codes. This construction involves multiple layers of binary codes, where a code in one layer is the square of the code in the previous layer. We also present a sufficient condition that allows checking nonlinearity of the $\mathbb{Z}_{2^L}$-linear codes by simple binary operations in their respective associated codes. Finally, we employ our arguments to verify the linearity of well-known $\mathbb{Z}_{2^L}$-linear code constructions, including the Hadamard, simplex, and MacDonald codes.

cs.IT↗