有向烛台形四元系的构造

发布时间：2018-12-16 08:25

【摘要】：一个阶数为v,指标为λ的有向烛台形t-设计DCSλ(t,K,v)是一个四元组(X,S,g,B),其中x是一个v元集；S是x的一个s元子集(称作干)；g是由x\S的一些非空子集构成的集合(其元素称作组),且划分X\S;召是X的某些有向子集构成的集合(其元素称作区组),且对任意B∈B,有|B|∈K；并且x中任意一个有向t-子集T,如果对每个i,|T∩(S∪Gi)|t,那么T恰好包含于召的λ个区组中；而对任意i,S ∪ Gi的任意有向t-子集都不包含于召中任何区组中.若g有ni个大小为gi的组(1≤i≤r),并且干的大小为s,则记其型为(91n1 92n2…grnr:s)当t=3,K={4}时,常称其为指标为λ的有向烛台形四元系,记作DCQSλ(g1n1 g2n2…grnr:s).有向烛台形设计是烛台形设计的推广形式.它在构作有向t-设计时起重要作用.本文利用递归构作与直接构作相结合的方法讨论了有向烛台形四元系的存在性问题,给出了如下一些结果：(1)DCQSλ(gn:0)存在的充要条件为λg(n-1)三0(mod 2),λgn≡0(mod 2),λg2n(n-1)[g(n+1)-3]≡0(mod 4),n≥3,g≥1.(2)DCQSλ(g2:0)存在的充要条件为λg≡0(mod 2),g≥2.(3)DCQSλ(g4:s)存在的充要条件为λg≡0(mod 2),λs≡0(mod 2),0≤s≤ 2g,g≥1.(4)当λ,g,s满足下列条件时,存在DCQSλ(g3:s).(a)λ≡1(mod 2),g≡0(mod 2),s≡0(mod 2),0≤s≤g,g≥2;(b)λ≡0(mod 2),0≤s≤g,g≥1.(5)当λ,g,s满足下列条件时,存在DCQSλ(g5:s).(a)λ≡1(mod 2),g≡0(mod 2),s≡0(mod 2),0≤s≤3g,g≥2;(b)λ≡0(mod 2),0≤s≤3g,g≥1.
[Abstract]:A directed candlestick t- design DCS 位 (tn Knv) with order v and index 位 is a quaternion (XG), where x is a set of v elements, S is a subset of s elements of x (called dry). G is a set composed of some nonempty subsets of x\ S (whose elements are called groups) and is divided into X\ S.The calling is a set of some directed subsets of X (whose elements are called block groups), and for any B 鈭，

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