On the non-existence of maximal difference matrices of deficiency 1

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.