Generalized Howell Designs With Block Size Four

Combinatorial design is an important part of combinatorial mathematics,which is specialized in studying the existence of various configurations and their structures.It has a long history to study combinatorial designs on square arrays,which has important application rules in coding theory.A generalized Howell design is both a class of doubly resolvable packing designs and a kind of combinatorial designs on square array.Meanwhile,it generalizes the Howell design and the Kirkman square,whereas the existence of Room squares,Howell designs and Kirkman squares has already been solved.There are few known results on generalized Howell designs,which are mainly concentrated on the cases of k = 2,3.In 2020,J.Shi et al.gave the specific structures of v = 88,s = 22,k = 4 and determined the existence of generalized Howell designs of v = 4n,s = n,k = 4,combining with the existence of four mutually orthogonal Latin squares of order n.In this paper,we will study the construction,existence and application of generalized Howell designs with k = 4.This paper is divided into five chapters:In chapter 1,we give some related concepts and existing results of generalized Howell designs.In chapter 2,the definitions of generalized Howell frame,incomplete generalized Howell design and other related auxiliary designs are given,and some recursive constructions are established by using these auxiliary designs.In chapter 3,using the recursive construction established in the second chapter and combining with the direct construction of small parameter designs,some new classes of generalized Howell designs are constructed.In chapter 4,taking advantage of the relationship between generalized Howell design,generalized packing design and multiply constant-weight code,some kinds of the optimal multiple constant weight codes with weight ! = 6 and the minimum distance d = 10 are obtained.In addtion,we generalize multiple constant weight codes and obtain some kinds of two dimensional multiple constant-weight codes.In chapter 5,we summarize the main conclusions of the paper and further research problems.