Paul C. Leopardi | Searcharxiv

arXiv2019

Gastineau-Hills' quasi-Clifford algebras and plug-in constructions for Hadamard matrices

The quasi-Clifford algebras, and their Wedderburn structure and representation theory, as described by Gastineau-Hills in 1980 and 1982, should be better known, and have only recently been rediscovered. These algebras and their representation theory provide effective tools to address certain questions relating to plug-in constructions for Hadamard matrices. The key question addressed is: Given $λ$, a pattern of amicability / anti-amicability, with $λ_{j,k}=λ_{k,j}=\pm 1$, find a set of $n$ monomial $\{-1,0,1\}$ matrices $\{D_j\}$ of minimal order such that $$ D_j D_k^T - λ_{j,k} D_k D_j^T = 0 \quad (j \neq k). $$

math.CO↗

arXiv2015

Twin bent functions and Clifford algebras

This paper examines a pair of bent functions on $\mathbb{Z}_2^{2m}$ and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical basis for the real representation of the Clifford algebra $\mathbb{R}_{m,m}.$ Some other necessary conditions are also briefly examined.

math.CO↗