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

Abstract

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