Construction of 2-balanced (n,k,λ) arrays

Frank Kwang-Ming Hwang and Shen Lin

Vol. 64 (1976), No. 2, 437–453

DOI: 10.2140/pjm.1976.64.437

Abstract

Let I_n denote the set of positive integers {1,2,⋯,n}, and I_n^λ the multiset consisting of λ copies of I_n. A submultiset S of I_n^λ is t-balanced if S can be partitioned into t parts such that the sums of all elements in each part are all equal. A t-balance (n,k,λ) array is a partition of I_n^λ into m multisets S_i, i = 1,⋯,m, which are all of size k and t-balanced. In this paper, we give a necessary and sufficient condition for the existence of 2-balanced (n,k,λ) arrays. Furthermore, we show how 2-balanced (n,k,λ) arrays can be used to construct a class of neighbor designs used in serology, or to give coverings of complete multigraphs by k-cycles.

Mathematical Subject Classification 2000

Primary: 05B30

Milestones

Received: 1 March 1974

Published: 1 June 1976