## Mutually orthogonal binary frequency squares

**Author: **Britz, Thomas; Cavenagh, Nicholas J.; Mammoliti, Adam; Wanless, Ian M.

**Date:** 2020

**Publisher: **The Electronic Journal of Combinatorics

**Type: **Journal article

**Link to this item using this URL: **https://hdl.handle.net/10289/13697

### Abstract

A frequency square is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only binary frequency squares of order n with n/2 zeros and n/2 ones in each row and column. Two such frequency squares are orthogonal if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a set of k-MOFS (n) is a set of k binary frequency squares of order n in which each pair of squares is orthogonal. A set of k-MOFS (n) must satisfy k≤(n−1)², and any set of MOFS achieving this bound is said to be complete. For any n for which there exists a Hadamard matrix of order n we show that there exists at least 2 ⁿ²/⁴−ᴼ ⁽ⁿˡᵒᵍⁿ ⁾ isomorphism classes of complete sets of MOFS (n). For 2

**Citation:** ["Britz, T., Cavenagh, N. J., Mammoliti, A., & Wanless, I. M. (2020). Mutually orthogonal binary frequency squares. The Electronic Journal of Combinatorics, 27(3). https://doi.org/10.37236/9373"]

**Copyright: **© The authors. Released under the CC BY-ND license (International 4.0).