Expanding generalized Hadamard matrices over Gm by substituting several generalized Hadamard matrices over G

Over an additive abelian group G of order g and for a given positive integer λ, a generalized Hadamard matrix GH(g, λ) is defined as a gλ × gλ matrix [h(i, j)], where 1 ≤ i ≤ gλ and 1 ≤ j· ≤ gλ, such that every element of G appears exactly λ times in the list h{i1, 1) − h(i2,1), h(i1, 2) − h(i2, 2), …, h(i1, gλ) − h(i2, gλ), for any i1 ≠ i2. In this paper, we propose a new method of expanding a GH(g^m, λ1) = B = [-Bij] over G^m by replacing each of its m-tuple Bij with Bij ⊕ GH(g, λ2) where m = gλ2. We may use g^m λ1 (not necessarily all distinct) GH(g, λ2)'s for the substitution and the resulting matrix is defined over the group of order g.