Difference matrices related to Sophie Germain primes $$p$$ p using functions on the fields $$F_{2p+1}$$ F 2 p + 1

Abstract

A $$k\times u\lambda $$ k × u ? matrix $$M=[d_{ij}]$$ M = [ d i j ] with entries from a group $$U$$ U of order $$u$$ u is called a $$(u,k,\lambda )$$ ( u , k , ? ) -difference matrix over $$U$$ U if the list of quotients $$d_{i\ell }{d_{j\ell }}^{-1}, 1 \le \ell \le u\lambda $$ d i l d j l - 1 , 1 ≤ l ≤ u ? , contains each element of $$U$$ U exactly $$\lambda $$ ? times for all $$i\ne j$$ i ? j . D. Jungnickel has shown that $$k \le u\lambda $$ k ≤ u ? . However, no general method is known for constructing difference matrices with arbitrary parameters. In this article we consider the case that the parameter $$u=p (>2)$$ u = p ( > 2 ) is a quasi Sophie Germain prime, where $$\lambda =q (=2p+1)$$ ? = q ( = 2 p + 1 ) is a prime power, and show that there exists a $$(p,(p-1)q/2,q)$$ ( p , ( p - 1 ) q / 2 , q ) -difference matrix over $${\mathbb {Z}}_p$$ Z p using functions from $$F_q$$ F q to $$F_q\setminus \{0\}$$ F q \ { 0 } . Our method is to construct a dual of TD $$_q((p-1)q/2,p)$$ q ( ( p - 1 ) q / 2 , p ) by using a group of order $$p^2q$$ p 2 q which acts regularly on the set of points but not on the set of blocks.