On Generalized Perfect Codes and Steiner Systems

摘要

The set of N-sequences of integers mod 2, i.e., (Z sub 2) to the nth power = A, is considered. For any x and y in A, the Hamming distance, H (x,y), counts the number of their different coordinates, i.e. H(x,y) = (the absolute value i, x sub i does not = y sub i), and is the weight of x-y, i.e., H(x,y) = H(x-y, the all zero element of A) = W(x-y). The term x is identified with its support, i.e., supp (x) = i, x sub i = 1, and A with the collection of subsets of 1,2,...,N. In this case, H(x,y) = (the absolute value of (x symmetric difference Y)) and W(x) = (the absolute value of x). THe Hamming sphere of radius r centered on x is expressed. A partition of A with N sub i Hamming spheres of radius r sub i is denoted by P(N, r sub i, N sub i, i = 1,2, . . .,s). When s = 1, the centers of the spheres form a perfect r sub 1 error correcting code (N, r sub 1 = 2 to the kth power). A generalized t-Steiner system is written GS(t,K = K sub 1, . . .,K sub s, N), i.e., a set of n points and a set of blocks with cardinalities in K so that any t-tuple of points belongs to exactly one block. For s = 1, this is the classical t-Steiner system, written S(t, k, n).