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01.06.2015 | Ausgabe 3/2015

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

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

## 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.

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