The existence of r‐golf designs

Abstract

An r‐golf design of order v, briefly by r‐G(v), is a large set of idempotent Latin squares of order v (ILS ( v )s) which contains r symmetric ILS ( v )s and v − r − 2 2 transposed pairs of ILS ( v )s. In this paper, we mainly consider the existence problem of r‐G(v)s. We present several recursive constructions and also display some direct constructions. As an application, several infinite classes of r‐G(v)s are determined, including the existence of 0‐G(v)s for more than half of admissible parameters and the existence of r‐G(v)s where v ≡ 110.3em ( mod0.3em 24 ) and r is any admissible integer ( r odd and 1 ≤ r ≤ v − 2).