Expanding generalized Hadamard matrices over G^m 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 x 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 = [B_ij] over G^m by replacing each of its m-tuple B_ij with B_ij ⊕ 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.