Electronic Notes in Discrete Mathematics 26 (2006) 37–38 www.elsevier.com/locate/endm
On Two Classes of Menon Designs Dean Crnkovi´c 1 Department of Mathematics Faculty of Philosophy University of Rijeka Rijeka, Croatia
Abstract Let p and 2p − 1 be prime powers and p ≡ 3 (mod 4). Then there exists a symmetric design with parameters (4p2 , 2p2 − p, p2 − p). Thus there exists a regular Hadamard matrix of order 4p2 . In a similar way we construct a symmetric (4(p+1)2 , 2p2 +3p+ 1, p2 + p) design D when p and 2p + 3 are prime powers, such that p ≡ 3 (mod 4). Keywords: regular Hadamard matrix, symmetric design, Menon design.
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Introduction
A 2-(v, k, λ) design is a finite incidence structure (P, B, I), where P and B are disjoint sets and I ⊆ P × B, with the following properties: 1. |P| = v; 2. every element of B is incident with exactly k elements of P; 3. every pair of distinct elements of P is incident with exactly λ elements of B. 1
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D. Crnkovi´c / Electronic Notes in Discrete Mathematics 26 (2006) 37–38
The elements of the set P are called points and the elements of the set B are called blocks. If |P| = |B| = v and 2 ≤ k ≤ v − 2, then a 2-(v, k, λ) design is called a symmetric design. A Hadamard matrix of order m is an (m × m)-matrix H = (hi,j ), hi,j ∈ {−1, 1}, satisfying HH T = H T H = mI, where I is the identity matrix. A Hadamard matrix is regular if the row and column sums are constant. The existence of a symmetric (4u2, 2u2 − u, u2 − u) design is equivalent to the existence of a regular Hadamard matrix of order 4u2 (see [2, Theorem 1.4 pp. 280]). Such symmetric designs are called Menon designs.
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Two Classes of Menon Designs
In [1] we prove the following assertion Theorem 2.1 Let p and 2p − 1 be prime powers and p ≡ 3 (mod 4). Then there exists a symmetric design with parameters (4p2 , 2p2 − p, p2 − p). When p and 2p − 1 are primes, the described Menon designs yield cyclic derived designs. Corollary 2.2 Let p and 2p − 1 be primes and p ≡ 3 (mod 4). Then there exists a cyclic 2-(2p2 − p, p2 − p, p2 − p − 1) design having an automorphism group isomorphic to (Zp : Z p−1 ) × (Z2p−1 : Zp−1). 2
In a similar way we construct a Menon design with parameters (4(p + 1) , 2p2 +3p+1, p2 +p) when p and 2p+3 are prime powers and p ≡ 3 (mod 4). These Menon designs produce 1-rotational derived designs. 2
Theorem 2.3 Let p and 2p + 3 be primes and p ≡ 3 (mod 4). There exists a 1-rotational 2-(2p2 + 3p + 1, p2 + p, p2 + p − 1) design having an automorphism group isomorphic to (Zp : Z p−1 ) × (Z2p+3 : Zp+1). 2
References [1] Crnkovi´c, D., A Series of Regular Hadamard Matrices, Des. Codes Cryptogr. 39, No. 2 (2006), 247–251. [2] Wallis, W. D., A. P. Street, and J. S. Wallis, “Combinatorics: Room Squares, Sum-Free Sets, Hadamard matrices”, Springer-Verlag, Berlin-Heidelberg-New York, 1972.