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Improving two theorems of bose on difference families

✍ Scribed by Marco Buratti

Publisher: John Wiley and Sons
Year: 1995
Tongue: English
Weight: 485 KB
Volume: 3
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.3180030104

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✦ Synopsis

In [2] R. C. Bose gives a sufficient condition for the existence of a (q, 5, 1) difference family in (GF(q), +)-where q = 1 mod 20 is a prime power-with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a (q,4,1) difference family in (GF(q, +)-where q = 1 mod 12 is a prime power-with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2-designs with block sizes 4 and 5. In particular, we get a ( P ~~, S , ~) -D F for any odd prime p = 2,3 (mod 5), and a (pZn,4, 1)-DF for any odd prime p = 2 (mod 3). 0 1995 John Wiley & Sons, Inc.

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