On Sylvester-type constructions of Hadamard matrices and their modifications

TL;DR

This paper introduces a modified Sylvester-type construction for Hadamard matrices using concatenation ideas, enabling the creation of larger matrices with varied ranks and kernels from existing collections.

Contribution

It presents a novel construction method for Hadamard matrices based on two collections, expanding possibilities for matrices with different algebraic properties.

Findings

Constructed Hadamard matrices of order $km$ from existing matrices.

Generated matrices with varied ranks and kernel dimensions.

Provided a flexible framework for new Hadamard matrix constructions.

Abstract

Using the ideas of concatenation construction of codes over the $q$-ary alphabet, we modify the known generalized Sylvester-type construction of the Hadamard matrices. The new construction is based on two collections of the Hadamard matrices. In particular this construction involves $m$ Hadamard matrices of order $k$ and $k$ Hadamard matrices of order $m$. These matrices are not necessary different. As a result we obtain a Hadamard matrix of order $km$. The new construction gives many possibilities for construction of the new Hadamard matrices with different ranks and dimension of kernel.