Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order
type: Item
instance of: scholarly article
title: Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order
zbMATH Open document ID: 0638.05010
DOI: 10.1016/0012-365X(88)90179-3
published in: Discrete Mathematics
publication date: 1988-01-01T00:00:00Z
review text: An F(n;\(\lambda\),...,\(\lambda)\) frequency square is an \(n\times n\) array consisting of the numbers 1,2,...,m with \(\lambda m=n\), such that each of the numbers 1,2,...,m occurs exactly \(\lambda\) times in each row and in each column. Two such squares are orthogonal if when superimposed, each ordered pair (i,j) with \(1\leq i\), \(j\leq m\) occurs exactly \(\lambda\) 2 times and a set of such squares is complete if the set contains (n-1) 2/(m-1) mutually orthogonal squares. Let p be a prime and let i divide s where s is a positive integer. In this paper the author constructs a complete set of (p 2-1) \(2/(p^{s/i}-1)\) mutually orthogonal F(p s;\(\lambda\),...,\(\lambda)\) frequency squares where \(\lambda =p^{(i- 1)s/i}\). If \(GF(p^{s/i})\) denotes the finite field of order \(p^{s/i}\), the proof is obtained by viewing the squares as polynomials over \(GF(p^{s/i})\) and by using elementary properties of orthogonal systems of polynomials over finite fields.
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 4039921
zbMATH Keywords: frequency square
zbMATH Keywords: mutually orthogonal squares
zbMATH Keywords: finite field
author: Gary L. Mullen
reviewed by: Gary L. Mullen
MaRDI profile type: Publication
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Wikidata QID: Q126844093
description: scientific article; zbMATH DE number 4039921