On The Clique Distribution And Maximum Independent Sets Of The Subgraphs Induced By Binary Constant Weight Codes In Powers Of Hypercubes

In modern communication system,the code distance constraint is a basic starting point,and the design of a check matrix with clear rules based on the code distance is the key to further improve the efficiency of coding and decoding,as well as the key to improve the performance of codes.Constant weight codes are a kind of typical structural codes,thus,a basic problem in coding theory are to find the values of A(n,d,w),where A(n,d,w)is the value of the n-dimensional maximum code set with code weight of all codewords is w,and the Hamming distance d.Many famous scholars,such as Hamming and Vardy,have conducted in-depth researches,but the problem have not been completely solved.On this basis,this paper,from the perspective of graph theory,has done some related exploration for the issue.First of all,this paper regards A(n,d,w)as the size of maximum independent set of Qnd-1,w which is the subgraph of Qnd induced by some vertices,where Qnd-1 is the d-1thpower of n-dimensional hypercube.Thus,the problem of finding the value of A(n,d,w)is transformed into the problem of finding the maximum independent set of Qn(d-1,w).In order to study the maximum independent set of Qn(d,w),we analyzed the distribution and structural characteristics of its maximum clique from the perspective of structure.According to the structural characteristics of Qn(d,w)we give a distance partition of Qn(d,w)relative to the vertex(1,1,…,1,0,0,…,0)to obtain different distance sets,and then construct the largest subset of these distance sets satisfying a certain distance condition.On this basis,the number,structural characteristics and distribution of the maximum cliques in An(d,w)for 4≤d≤5 are completely obtained.Secondly,the method of constructing the maximum independent set is used to prove the characterizations of A(n,d,w)=2 and 3,and makes a preliminary exploration on the case of A(n,d,w)=4.