# Mathematics Seminar: Mutually Orthogonal Frequency Rectangles

**2 Feb 2023 3:10 PM - 4:10 PM**
**Presenter/Speaker:** Fahim Rahim, Department of Mathematics, Waikato University
**Location:** G.3.33

A *frequency rectangle* of type FR(*m*, *n*; *q*) is an *m* × *n* matrix such that each symbol from a set of size *q* appears *n/q* times in each row and *m/q* times in each column. Two frequency rectangles of the same type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of *k* frequency rectangles in which every pair is orthogonal is called a set of *mutually orthogonal frequency rectangles*, denoted by *k*–MOFR(*m*, *n*; *q*).

In this talk, I will discuss the application of frequency rectangles in experimental designs and their connection with orthogonal arrays and Hadamard matrices. I will also describe a stronger form of orthogonality for a set of frequency rectangles. A *k*–MOFR(*m*, *n*; *q*) is said to be* t–orthogonal*, if each subset of size *t*, when superimposed, contains each of the *q ^{t}* possible ordered

*t*-tuples of entries exactly

*mn/q*times. A set of vectors over a finite field F

^{t}_{q}is said to be

*t-independent*if each subset of size

*t*is linearly independent.

I will discuss a method to obtain a set of

*t*–orthogonal

*k*–MOFR(

*q*,

^{M}*q*,

^{N}*q*) corresponding to a set of

*t*–independent vectors in (F

*)*

_{q}*. Then I will talk about some bounds and known values in the literature on the size of a set of*

^{M+N}*t*–independent vectors.

A frequency rectangle of type FR(*n*, *n*; *q*) is called a frequency square. For *p* an odd prime, I will describe a construction for a set of (*p* − 1) binary MOFS of order 2*p*. This improves the lower bounds in (Britz et al. 2020) for the previously known values for *p* ≥ 19.