Combinatorial Constructions For Optimal Two-dimensional Optical Orthogonal Codes With The Best Cross-correlation Constraint

Optical orthogonal codes are commonly used as signature codes for optical code-division multiple access（OCDMA）systems.An optical orthogonal code is a family of sequences with good auto-correlation and cross-correlation properties.The code-division multiple access（CDMA）technique has been successfully used in satellite com-munication and mobile communication field.But due to the limitation of bandwidth,CDMA technique can’t express its feature greatly.The optical code-division multiple access system had solved this problem by combining the bandwidth resource with CD-MA technology.In order to further improve CDMA system performance,G.C.Yang proposed the concept of two-dimensional optical orthogonal code（2-D OOC）.A two-dimensional（n ×m,k,λ_a,λ_c）optical orthogonal code（briefly 2-D（n ×m,k,λ_a,λ_c）-OOC）,C,is a family of n ×m（0,1）-matrices（called codewords）of Ham-ming weight k satisfying the following two properties:（1）the autocorrelation property:for each matrix A =(a_ij)_n×m∈ C and each integer r,r（?）0（mod m）,（?）a_ija_i,j+r≤λ_a;（2）the cross-correlation property:for each matrix A =(a_ij)_n×m ∈ C,B =（bij）n×xm ∈C with A ≠ B,and each integer r,（?）aijbi,j+r≤λ_c,where the arithmetic j + r is reduced modulo m.This paper focuses on optimal two-dimensional optical orthogonal codes with the auto-correlation λ_a and the best cross-correlation 1.By examining the structures of w-cyclic group divisible designs and semi-cyclic incomplete holey group divisible designs,we present new combinatorial constructions for two-dimensional（n×m,k,λ_a,1）-optical orthogonal codes.As a result,when k = 3 and λ_a = 2,the exact number of codewords of an optimal two-dimensional（n ×m,3,2,1）-optical orthogonal code is determined for any positive integers n and m≡2（mod 4）.The organization of this thesis is as follows:In Chapter 1,we gives a brief introduction on the background of CDMA system,the concept of two-dimensional optical orthogonal code and research status of it.In Chapter 2,we need more g-regular 2-D OOCs,which can be derived from semi-cyclic holey group divisible designs.By examining the structures of w-cyclic group divisible designs and semi-cyclic incomplete holey group divisible designs,we present new combinatorial constructions for 2-D（[n:r]× m,k,λ_a,1）-OOC.In Chapter 3,when n ∈ {4,5,7,8,11},we are devoted to constructing optimal 2-D（n × m,k,λ_a,1）-OOC.In Chapter 4,we prove our main result which determines the exact value of Φ（n x m,3,2,1）and propose the problems of further research.