Schur Ring over Group $\Z_{2}^{n}$, Circulant $S-$Sets Invariant by Decimation and Hadamard Matrices

TL;DR

This paper explores the structure of Schur rings and $S$-sets over $ ext{Z}_2^n$, their invariance under decimation, and their relation to circulant Hadamard matrices, providing new insights into their existence and properties.

Contribution

It introduces the complete $S$-sets, studies their structure over $ ext{Z}_2^n$, and establishes their invariance under decimation, linking them to the existence of Hadamard matrices.

Findings

All $S$-sets are invariant under decimation.

If a Hadamard matrix exists, it is contained in a complete $S$-set.

Circulant Hadamard matrices with certain structures cannot exist.

Abstract

In this paper a variety of issues are discussed, Schur ring, $S$-sets, circulant orbits, decimation operator and Hadamard matrices and their relation between them is shown. Firstly we define the complete $S$-sets. Next, we study the structure of Schur ring with circulant basic sets over $\Z_{2}^{n}$ and we define the free and non-free circulant $S$-sets, the symmetric, non-symmetric and antisymmetric circulant $S$-sets. We prove that all this $S$-sets are invariants under decimation. Finally, we prove that if a Hadamard matrix exist then this is contained in a complete $S$-set. Also, we prove that can't exist circulant and with one core Hadamard matrices with some particular structure. These theorems include a result known on symmetric circulant Hadamard matrices of order $4n$ only when $n$ is an odd number.