【作者】 杜娟；

【导师】 王金华；

【作者基本信息】 南通大学 ， 应用数学， 2014， 硕士

【摘要】 设k,λ,r,和u为正整数.区组大小为k,指数为入,重复数为r,元素个数为u的广义Kirkman方,GKSk(u;1,λ;r),是定义在u元点集V上且满足下列条件的r×r的方阵S.(1)S的每一个单元要么为空,要么包含V的一个k元子集；(2)V中的每一个元在S的每一行、每一列中仅出现一次；(3)V的任意无序二元点对至多包含在S的λ个k元子集中.广义Kirkman方GKS2(u;1,1;r)就是Howell设计H(r,u).GKS2(v;1,1;u-1)是u-1阶的Room方.显然Howell设计是Room方的推广,广义Kirkman方是Howell设计的推广.Kirkman方就是r=k-4/λ(v-1)时的广义Kirkman方.Kirkman方记为KSk(u;1,λ).众所周知,双重可分解的平衡不完全区组设计(u,k,A)-DRBIBD就是Kirkman方.广义Kirkman方与双重可分解的可分组设计有密切的关系,双重可分解的可分组设计是一类广义Kirkman方.关于广义Kirkman方的研究已经做了很多工作.1975年,Mullin和Wallis建立了Room方的谱系.Howell设计存在性问题于1984年由Stinson等完全解决.KS3(u;1,1)和KS3(u;1,2)的存在性问题直到2008年才由Abel等学者基本确定.2013年Abel等基本建立了双重可分解拟Kirkman三元系DRNKTS(u)(即GKS3(u;1,1;(u-2)/2))的谱系.本文进一步研究广义Kirkman方的构作.主要讨论GKS3(4u;1,1,2(u-1))和GKS3(6u;1,1,3(u-1))的存在性问题.GKS3(4u;1,1,2(u-1))和GKS3(6u;1,1,3(u-1))等价于双重可分解GDD,即型为4u和6u的3-DRGDD.第二章利用标准的"starter-adder"方法直接构作了小阶的3-DRGDD,为后面的递推工作奠定基础.第三章构作了一些新的Frame,总结了区组大小为3,型为2t,4t和6t的Frame的相关存在性结论.第四章,首先利用Frame、GDD和PBD等建立3-DRGDD的递推构作.然后在递推构作的基础上建立了两类广义Kirkman方的谱系.即当钍≥3且u≡0(mod 3)时,GKS3(4u;1,1,2(u-1))存在,其中有17种可能例外；当u≥3时,GKS3(6u;1,1,3(u-1))存在,其中有31种可能例外.第五章描述了广义Kirkman方和双重常重量码、常复合码之间的关系.作为广义Kirkman方的直接应用得到了相关的码类.第六章展望了进一步的研究工作.更多还原

【Abstract】 Let k, λ, r, and v be positive integers. A generalized Kirkman square with block size k, index λ, replication number r, and v elements, GKSk(v; 1, λ; r), is an r x r array S defined on a v-set V such that(1) each cell of S is either empty or contains a κ-set of V,(2) every element of V occurs once in each row and column of S,(3) each 2-subset of V is contained in at most λ κ-sets of S.A generalized Kirkman, GKS2(u,1,1; r)), is just a Howell design H(r, u). A GKS2(u; 1,1; u-1) is a Room square of order u-1. Obviously, a Howell design is a generalization of a Room square, and a generalized Kirkman square is a gener-alization of a Howell design. When r=(λ(u-1)/(k-1) a generalized Kirkman square is a Kirkman square, and denoted by KSk(u;1,λ). As everyone knows, a doubly resolv-able (v, k,λ)-BIBD is just a Kirkman square. There are tight connections between generalized Kirkman squares and doubly resolvable packing designs. Doubly re-solvable GDDs are one kind of generalized Kirkman squares. A lot of work was done about generalized Kirkman squares. In 1975, Mullin and Wallis established the spectrum of Room squares. The existence of Howell designs has been completely determined by Stinson et al. in 1984. In 2008, Abel et.al. established almost the spectrums of KSs(u; 1, 1)s and KS3(u; 1,2)s. In 2013, Abel et al. furthermore studied the problem for the existence of doubly resolvable nearly Kirkman triple systems DRNKTS(u)s (i.e., GKS3(u;1,1; (u-2)/2)s). In this paper, we are interested in the constructions of generalized Kirkman squares. We will discuss the existence of GKS3(4u; 1,1,2(u-1))s and GKS3(6u; 1,1,Z(u-1))s, which are doubly resolvable GDDs,3-DRGDDs of types 4u and 6u.Section 2 uses standard "starter-adder" method to construct directly some new 3-DRGDDs of types 4u and 6u with small order u, which are foundation for our later recursive constructions.Section 3 describes a few new constructions for frames and summarizes some known results on frames with block size 3 and types 2t,4t, and 6t.Section 4 first establishes the recursive constructions for 3-DRGDDs by using frames, GDDs and PBDs. Then applying these recursive constructions shows the ex-istence of two kinds generalized Kirkman squares. Namely, there exist a 3-DRGDD of type 4u for u≥3 and u=0 (mod 3) with 17 possible exceptions, and a 3-DRGDD of type 6u for u≥3 with 31 possible exceptions.In Section 5, we discuss the tight connections between generalized Kirkman squares and doubly constant weight codes, constant composition codes. As appli-cations of generalized Kirkman squares, we obtain some new classes of codes.Section 6 gives some concluding remarks and problems for further research.更多还原