Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order
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.
scientific article; zbMATH DE number 4039921
scientific article