All v(3, t)'s exist for 3t + 1 a prime

✍ Scribed by G. H. J. Van Rees

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 284 KB
• Volume: 3
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.3180030604

✦ Synopsis

The existence of a V(3, t ) , for any prime 3 t + l is proved constructively. A V(rn, t ) is equivalent to rn idempotent pairwise orthogonal Latin squares of order (rn+l)t + 1 with one hole of order t. 0 1995 John Wiley & Sons, he.

1. Introduction

For the basic definitions about Latin squares the reader is referred to Denes and Keedwell [4]. Let q = mt + 1 be a prime power and let CO be a multiplicative subgroup of GF(q){O} of order t and index m. Let the cosets of this group be CO, Cl, . . . , C,-l.

These are called the cyclotomic classes of GF(q) of index m. For q = mt + 1 a prime power, Wilson defined an A(m, t ) to be the matrix-minus-diagonal of order k = m + 2 with elements aij, i # j , from GF(q) satisfying the property that for 1 I j l < j z 5 k , the set {ai,, -aij, 11 5 i 5 k , i # j l , i # j z } is a system of distinct representatives of the cyclotomic classes CO, CI, . . . , C,-1 . Wilson [7] proved the following lemma.

Lemma 1.1. Let q = mt + 1 be a prime power. Ifthere exists a matrix-minus-diagonal A(m, t), then there exists a set of m idempotent pairwise orthogonal Latin squares of order ( m + 1)t + 1 with one hole of size t.

For q = mt + 1 a prime power, Mullin et al. defined a V ( m , t ) to be a vector (bl, bz, . . . , b,+l) with elements from GF(q) satisfying the property that for each element k in the set {1,2,. . . , m + l}, the set {bi -bjliE{1,2,. . . , m + l}{m + 2 -k } , j =