On generalized Hadamard matrices GH(q,q)'s and GH(q,q²)'s

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A matrix H=[dij] is a generalized Hadamard matrix of order uλ with entries from U which is a finite group of order u (for short GH(u,λ)) such that whenever i≠ ℓ the set {dijdℓ ⱼ⁻¹| 1≤ j≤ uλ} contains each element of U exactly λ times. In this paper, we construct GH(q,q)'s and GH(q,q²)'s over additive groups of finite fields GF(q)'s by using some sorts of functions.