On the variance of average distance of subsets in the Hamming space

Authors:
Fang-Wei Fu;San Ling;Chaoping Xing
Affiliations:
Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore 117508, Singapore;Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore;Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore
Venue:
Discrete Applied Mathematics
Year:
2005

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Abstract

Let V be a finite set with q distinct elements. For a subset C of Vn, denote var(C) the variance of the average Hamming distance of C. Let T (n, M; q) and R (n, M; q) denote the minimum and maximum variance of the average Hamming distance of subsets of Vn with cardinality M, respectively. In this paper, we study T(n, M; q) and R(n, M; q) for general q. Using methods from coding theory, we derive upper and lower bounds on var(C), which generalize and unify the bounds for the case q = 2. These bounds enable us to determine the exact value for T(n, M; q) and R(n, M; q) in several cases.