Research On Steiner Systems And Two Classes Of Isodual Codes Over Finite Rings

In this dissertation,we mainly study the counting problems of non-fullrank Steiner triple and quadruple systems over finite fields.Meanwhile,based on the coding theory on finite ring and finite field,two kinds of isodual codes are studied,and both of them can be regarded as the generalization of double circulant.Specific contents are given as follows:1?The p-rank of a Steiner triple system S is the dimension of the linear span of the set of characteristic vectors of blocks of S,over Fp.We derive a formula for the number of different Steiner triple systems of order v and given 2-rank r1,r1<v.Using the same method,we further give the formula for the number of Steiner quadruple systems of order v and given 2-rank r2,r2<v.Our results extend previous work on enumerating Steiner triple(quadruple)systems according to the rank of their codes,mainly by V.A.Zinoviev and D.V.Zinoviev for the binary case.2?In a recent work,Jungnickel,Magliveras,Tonchev and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems of order v and 3-rank v-k.where v=3k.We develop an approach to generalize this bound and estimate the number of isomorphism classes of STS(v)of 3-rank v-k-1 for an arbitrary v of form 3kT,where T is congruent to 1 or 3 modulo 6.3?Double polycirculant(DP)codes are introduced here as a generalization of double circulant codes.When the matrix of the polyshift is a companion matrix of a trinomial,we show that such a code is isodual,hence formally selfdual.Self-duality can only occur over F2 in the double circulant case.When p ? 2(mod 3)we obtain an infinite sequence of explicit irreducible trinomials over Fp,and further show that binary double polycirculant codes are asymptotically good.We investigate DP codes over Zpm for the homogeneous,Lee and Euclidean distance.The families of DP codes corresponding to the Hensel lift of above irreducible trinomials over Zpm are shown to be asymptotically good for the three distances considered.Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes over Fp.The codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes for the Lee distance,and amongst isodual codes for the Euclidean distance over Z4.4?Double Toeplitz(DT)codes are introduced here as another generalization of double circulant codes.We show that such a code is isodual,hence formally self-dual.Self-dual DT codes are characterized as double circulant or double negacirculant.Even DT binary codes are characterized as double circulants.Numerical examples obtained by exhaustive search show that the codes constructed have best-known minimum distance,up to one unit,amongst formally self-dual codes,and sometimes improve on the known values.Over F4 an explicit construction of DT codes,based on quadratic residues in a prime field,performs equally well.We show that DT codes are asymptotically good over Fq.Specifically,we construct DT codes arbitrarily close to the asymptotic Varshamov-Gilbert bound for codes of rate one half.