Optical orthogonal code (OOC) has good correlation properties. It has many important appli-cations in a fiber-optic code-division multiple access channel. In this paper, a combinatorial construction foroptimal (15p, 5, 1) optical orthogonal codes with p congruent to 1 modulo 4 and greater than 5 is given byapplying Weil's Theorem. From this, when v is a product of primes congruent to 1 modulo 4 and greater than5, an optimal (15v, 5, 1)-OOC can be obtained by applying a known recursive construction.
Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number,Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n. When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v=3. In this paper wewill show that Ushio Conjecture is true when v=4k-1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k-1)m≤2kn, (2) (2k-1)n ≤2km, (3) m+n ≡0 (mod 4k-1), (4) (4k-1)mn/[2(2k-1)(m+n)] is an integer.integer.
DU Beiliang & WANG Jian Department of Mathematics, Suzhou University, Suzhou 215006, China
Let Km,n be a completebipartite graph with two partite sets having m and n vertices,respectively. A Kp,q-factorization of Km,n is a set ofedge-disjoint Kp,q-factors of Km,n which partition theset of edges of Km,n. When p=1 and q is a prime number,Wang, in his paper 'On K1,k-factorizations of a completebipartite graph' (Discrete Math, 1994, 126: 359-364),investigated the K1,q-factorization of Km,n and gave asufficient condition for such a factorization to exist. In the paper'K1,k-factorizations of complete bipartite graphs' (DiscreteMath, 2002, 259: 301-306), Du and Wang extended Wang's resultto the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have aKp,q-factorization. As a special case, it is shown that theMartin's BAC conjecture is true when p:q=k:(k+1) for any positiveinteger k.
