指标为3的单纯可迁三元系大集的三倍构作

Tripling Construction for Large Sets of Directed Triple Systems with Index 3

一个指标为3的可迁三元系DTS(v,3)是一个对子(X,B)(其中X为一个v元集,B为X中可迁三元组(称作区组)的集合),满足X的每个有序对都恰包含于B中的3个区组。设(X,B)是一个没有重复区组的DTS(v,3),如果(x,y,z)∈B,必有(z,y,x),(z,x,y),(y,x,z),(y,z,x),(x,z,y)埸B,则称(X,B)是单纯的,记为PDTS(v,3)。不相交PDTS(v,3)大集记为LPDTS(v,3),是一个集合{(X,Bi)}i,其中每个(X,Bi)都是PDTS(v,3),并且∪iBi构成X中所有可迁三元组的一个划分。本文给出了LPDTS(v,3)的一种三倍构造方法,得到了其存在的一个无穷类:对于任意正整数v,v≡8,14(mod18),存在LPDTS(v,3)。结论对构作常重码具有重要的参考价值和理论意义。

There are two kinds of oriented triples on X：the cyclic triple and the transitive triple.A cyclic triple on X is a set of three ordered pairs（x,y）,（y,z） and（z,x） of X,which is denoted by x,y,z（or y,z,x,or z,x,y）,and a transitive triple on X is a set of three ordered pairs（x,y）,（y,z） and（x,z） of X,which is denoted by（x,y,z）.An oriented triple system of order v with index λ is a pair（X,B）where X is a v-set and B is a collection of oriented triples on X,called blocks,such that every ordered pair of X belongs to exactly λ blocks of B.If B consists of transitive（or cyclic） triples only,the system is called a directed triple system（or Mendelsohn triple system） of order v with index λ and denoted by DTS（v,λ）（or MTS（v,λ））.If B contains both cyclic triples and transitive triples,the system is called a hybrid triple system of order v with index λ and denoted by HTS（v,λ）.A triple system is called simple if there are no repeated blocks in B.A simple DTS（v,λ） is called pure and denoted by PDTS（v,λ） if（x,y,z）∈B implies（z,y,x）,（z,x,y）,（y,x,z）,（y,z,x）,（x,z,y）∈B.A large set of disjoint PDTS（v,λ）s,denoted by LPDTS（v,λ）,is a collection of {（X,Bi）}i where each（X,Bi） is a PDTS（v,λ） and ∪iBi is a partition of all transitive triples on X.In this paper,a tripling construction for LPDTS（v,3） is presented,and one infinite family for the existence of LPDTS（v,3） is obtained：for any positive integers v,v≡8,14（mod18）,there exists an LPDTS（v,3）.