On the non-existence of maximal difference matrices of deficiency 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 $i\ne j.$ Jungnickel has shown that $k \le u\lambda$ and it is conjectured that the equality holds only if $U$ is a $p$-group for a prime $p.$ On the other hand, Winterhof has shown that some known results on the non-existence of $(u,u\lambda ,\lambda )$-difference matrices are extended to $(u,u\lambda -1,\lambda )$-difference matrices. This fact suggests us that there is a close connection between these two cases. In this article we show that any $(u,u\lambda -1,\lambda )$-difference matrix over an abelian $p$-group can be extended to a $(u,u\lambda ,\lambda )$-difference matrix.