Combinatorial Constructions Of Optimal Cyclic Constant-weight Codes

Constructing various optimal codes is a fundamental problem in coding theory.Generally speaking,the problem can be divided into two parts.One part is to find bounds on the size of a code,including the upper bound and lower bound,and the other is to find appropriate constructions for the codes.Clearly,the problem is a challenging task.Even for a simple parameter,the construction of an optimal code is not a easy work.Cyclic constant-weight codes are of an important class of combinatorial codes.In the literature,the research on cyclic constant-weight codes is focused on binary codes.Nonbinary cyclic constant-weight codes are systematically studied for the first time in this thesis.We define cyclic constant-weight codes from the perspective of combinatorial de-signs,provide some new combinatorial constructions,and present a series of optimal nonbinary cyclic constant-weight codes.An(n,d,w)q code stands for a q-ary code of length n,(constant)weight w and(minimum)distance d.The thesis is organized as follows.In Chapter 1,we give a brief introduction to the application background of cyclic constant-weight codes and present the main results of this thesis.In Chapter 2,the set-theoretic definition and equivalent combinatorial descriptions of cyclic constant-weight codes are introduced.We then recall some related concep-tions and notations in combinatorial design theory,mostly including the pure and mixed difference method,cyclic packing designs and Skolem-type sequences.In Chapter 3,when the minimum distance d ? {2w-3,2w-2,2w-1},we give an upper bound on the size(the number of codewords)of a cyclic(n,d,w)q code by employing the pure and mixed difference method.When d ? {1,2} or d ? min{n,2w},the exact value of the size of an optimal cyclic(n,d,w)q code is completely determined.The exact value of the size of an optimal cyclic(n,5,3)q code is also determined.In Chapters 4 and 5,by using special sequences,especially the Skolem-type se-quences,we directly construct optimal cyclic(n,4,3)3 and(n,3,3)3 codes,respectively.For any positive integers n and d ? {3,4},the exact value of size of an optimal cyclic(n,d,3)3 code is completely determined.In Chapter 6,we give a new construction for optimal cyclic(n,4,3)4 codes.The construction depends on three strictly cyclic packings(SCPs)which are mutually orbit-disjoint and have given difference leaves.We directly construct the desired SCPs by using special sequences,and then completely determine the exact value of the size of an optimal cyclic(n,4,3)4 code for all n.In Chapter 7,we investigate cyclic(n,3,3)4 codes.By thoroughly analyzing the pure and mixed differences,we first give an improved the upper bound on the size of an optimal cyclic(n,3,3)4 code if n ? 2,4(mod 8).We then provide two constructions for optimal cyclic(n,3,3)4 codes.By employing special sequences,especially Skolem-type sequences,optimal cyclic(n,3,3)4 codes are explicitly constructed.For any n(?)18(mod 24),we determine the exact value of the size of an optimal cyclic(n,3,3)4 code.Chapter 8 is devoted to cyclic(n,6,4)3 codes.An upper bound on the size of a cyclic(n,6,4)3 code is given by studying the pure and mixed difference.A novel construction for cyclic(n,6,4)3 codes,based on two super orbit-disjoint(n,4,1)-SCPs,is provided.For every positive integer n(?)0,6,18(mod 24),we construct an optimal cyclic(n,6,4)3 code.In addition,some infinite classes of optimal cyclic(n,6,4)3 codes are given.In Chapter 9,a brief conclusion is given and the future research plans and prospects are introduced.