Equivalence classes of central semiregular relative difference sets

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.

Section: Quasi-orthogonal Cocyclesmentioning

confidence: 65%

“…However, both properties are preserved by a certain “shift action” on each cocycle class. For

a \in G

, this action maps

ψ \in Z^{2} false(G, Z_{2} false)

ψ a : = ψ \partial ψ_{a}

, where

ψ_{a} false(x false) = ψ false(a, x false)

; see [, Definition 3.3]. By Lemma , the sum

0true \sum_{h \in G} ψ (a, h) ψ (ag, h)

of row

g \neq 1

M_{ψ a}

is either a noninitial row sum of

M_{ψ}

, or the negation of one.…”

Section: Quasi‐orthogonal Cocyclesmentioning

confidence: 99%

On quasi‐orthogonal cocycles

Armario

Flannery

2017

A group extensions approach to relative difference sets

Galati¹

2004

A new approach to (normal) relative difference sets (RDSs) is presented and applied to give a new method for recursively constructing infinite families of semiregular RDSs. Our main result (Theorem 7.1) shows that any metabelian semiregular RDS gives rise to an infinite family of metabelian semiregular RDSs. The new method is applied to identify several new infinite families of non‐abelian semiregular RDSs, and new methods for constructing generalized Hadamard matrices are given. The techniques employed are derived from the general theory of group extensions. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 279–298, 2004.

Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles

LeBel

Horadam

2008

Self Cite

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non-cyclic abelian groups and use it to find all the orthogonal cocycles over Z t 2 , 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z t 2 , t ≥ 2, must be multiplicative.