Existence of r-self-orthogonal Latin squares

AbstractTwo Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the first one, we say that the first square is r-self-orthogonal, denoted by r-SOLS(v). It has been proved that for any integer v⩾28, there exists an r-SOLS(v) if and only if v⩽r⩽v2 and r∉{v+1,v2-1}. In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares

The existence of directed BIBDs

AbstractFor any positive integers k⩾3 and λ, let cd(k,λ) denote the smallest integer such that the necessary conditions 2λ(v−1)≡0(modk−1) and λv(v−1)≡0(mod(k2)) for the existence of a DB(k,λ;v) are also sufficient for every v⩾cd(k,λ). In this article we provide an estimate for cd(k,λ) when k≡0(mod4) and any λ. Combined with the results in (Discrete Math. 222 (2000) 27–40), we completely give an estimate of cd(k,λ) for any integers k⩾3 and λ