On quasi‐orthogonal cocycles

Abstract

We introduce the notion of quasi‐orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±1}‐matrices of size congruent to 2 modulo 4. Quasi‐orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi‐Hadamard groups, relative quasi‐difference sets, and certain partially balanced incomplete block designs, are proved.

Generalized binary arrays from quasi-orthogonal cocycles

“…The next result is mostly Proposition 4.3 in [3]. For each ψ ∈ Z 2 (G, Z 2 ) we have a canonical central extension E ψ with element set {(±1, g) | g ∈ G} and multiplication defined by (u, g)(v, h) = (uvψ(g, h), gh).…”

“…In analogy with the definition of orthogonal cocycles, we say that ψ is quasiorthogonal if its matrix has least possible row excess: by Proposition 1, either ψ ∈ B 2 (G, Z 2 ) and RE(M ψ ) = 4t, or ψ ∈ B 2 (G, Z 2 ) and RE(M ψ ) = 8t + 2 (coboundaries were excluded from the notion of quasi-orthogonality in [3]).…”

“…Many kinds of pairwise combinatorial designs are normal matrices (the defining pairwise constraint on rows implies the same constraint on columns; see [7,Chapter 7]). We also note that the matrix of a quasi-orthogonal cocycle is normal [3,Remark 6]. Thus, by the following lemma derived from (2), a cocycle ψ is quasi-orthogonal if and only if M ψ has optimal column excess.…”

“…Just as the notion of GPBA extends that of GPBS to dimensions greater than 1, a GOBA (generalized optimal binary array) is a higher-dimensional version of a GOBS. Section 3 treats GPBAs and GOBAs from the perspective of [3]. We prove a one-toone correspondence between GOBAs, quasi-orthogonal cocycles over abelian groups, and abelian relative quasi-difference sets.…”

Almost supplementary difference sets and quaternary sequences with optimal autocorrelation

“…We explain how to characterize quaternary sequences of odd length n with optimal autocorrelation as almost supplementary difference sets in Z n . This paper is a natural successor to [4,5], which initiated the theory of quasi-orthogonal cocycles and their applications in design theory. We obtain new existence results for such cocycles from a connection to optimal quaternary sequences.…”