Research On Steiner Triple System And Trace Codes Over Finite Rings

In this thesis,the structural properties of a Steiner triple system and its isomorphic classes are studied.Meanwhile,based on the coding theory on finite rings,trace codes on a finite ring are studied.1、We prove several structural properties of the Steiner triple systems(STS)of order 3w+3 that include one or more transversal subdesigns TD(3,w).Using an exhaustive search,we find that there are 2004720 isomorphism classes of STS(21)including a TD(3,6)(or,equivalently,a 6-by-6 latin square).2、We construct two new infinite families of trace codes of dimension 2m,over the ring IFp+uFp,with u2=when p is an odd prime.They have the algebraic structure of abelian codes.We calculate Lee weight distributions of these trace codes by Gauss sums.By Gray mapping,we obtain two infinite families of linear p-ary codes of respective lengths(pm-1)2 and 2(pm-1)2.When m is singly-even,the first family gives five-weight codes.When m is odd,and p≡3(mod 4),the first family yields p-ary two-weight codes,which are shown to be optimal by application of the Griesmer bound.The second family consists of two-weight codes that are shown to be optimal,by the Griesmer bound,whenever p=3 and m>3,or p>5 and m>4.Applications to secret sharing schemes are given.3、Two families of few-weight codes for the Lee weight over the ring Rk,based on two different defining sets7 are constructed.For the first defining set,taking the Gray map,we obtain an infinite family of binary two-weight codes which are in fact 2k-fold replicated MacDonald codes.For the second defining set,a family of three-weight codes is established.All the nonzero codewords of these few-weight codes are minimal under certain conditions.