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01.06.2015 | Ausgabe 3/2015

Designs, Codes and Cryptography 3/2015

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

Designs, Codes and Cryptography > Ausgabe 3/2015
Yutaka Hiramine, Chihiro Suetake
Wichtige Hinweise
Communicated by D. Jungnickel.


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.

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