The Existence Of G-OD_?(3,4,v)

An ordered design OD?(t,k,v) is a ?(?)t!×k array, where |x| =v, satisfying that every ?(?)t! × t subarray contains each ordered t-subset of X exactly ? times. If an OD?(t, k, v) has a conjugate invariant subgroup G, then it is denoted by G-OD?(t, k, v).An OD?(3,4,v) is an ordered generalization of a QS(v,?). Specially, an OD(3,4, v)is closely related to an OA(3,4,v), and equivalent to a 2-idempotent 3-quasigroup.OD?(3,4,v) has important applications in the fileds of computer science and cryptog-raphy. In this paper, by direct and recursive constructions, we mainly investigate the existence of G-OD?(3,4,v)? and get the following results:(1) We determine the existence of G-OD?(3,4,v) for G?C3,S3 and K4. Combin-ing with some known results, we almostly give the existence of G?OD?(3,4,v) for G a subgroup of S4.(2) When v?0 (mod 4), we determine the existence of the resolvable ordered design with conjugate invariant subgroup D4 (D4-ROD(3,4,v)). By the relationship of the subgroups of S4, we can determine some existence of G-ROD(3, 4, v) for G a subgroup of S4.