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

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.