On The Construction Of Generalized Kirkman Squares

In recent years, constant weight codes are introduced in a number of engineering appli-cations, including code-division multiple access (CDMA) systems of optical fibers, frequency-hopping spred-spectrum systems, radar and sonar signal designs, mobile radio and synchroniza-tion. An (n,d,ω) constant weight binary code is a set of binary vectors of length, such thateach vector contains ones and n-ω zeros, and any two vectors differ in at least positions(always called Hamming distance). Doubly constant weight codes is useful in developing theupper bounds on (n,d,ω). T. Etzion pointed out that several combinatorial designs, such asSteiner systems, Kirkman squares etc., are tightly connected with doubly constant weight codes.Permutation arrays are of recent interest because of their applications to data transmissionover power lines. Deza M., Vanstone S. A. pointed out that the generalized Kirkman squaresis tightly connected with permutation arrays.This paper mainly discuss the existence of generalized Kirkman squares GKS(n+t,3n),when t>1. It is divided into four parts. In the first part, we mainly introduce the backgroundof Kirkman square and its definition. Then we list the known results and our paper’s main con-clusions. In the second part, we give the definitions and results of some auxiliary designs. In thethird part, we give the Intransitive Starter-Adder method and some recursive constructions.In the fourth part, the main conclusions are given by using the recursive constructions and theexistence of some small orders.