# On quasi-orthogonal cocycles

## TL;DR

This paper introduces quasi-orthogonal cocycles motivated by the maximal determinant problem for certain square matrices, establishing their connections to various combinatorial objects and extending algebraic design theory.

## Contribution

It defines quasi-orthogonal cocycles and proves their equivalences with several known and new combinatorial structures, expanding the theoretical framework.

## Key findings

- Established the concept of quasi-orthogonal cocycles.
- Proved equivalences with quasi-Hadamard groups and relative quasi-difference sets.
- Linked quasi-orthogonal cocycles to partially balanced incomplete block designs.

## Abstract

We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square $\{\pm 1\}$-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.