On generalized Hadamard matrices GH(q,q)'s and GH(q,q^2)'s

A matrix $H=[d_{ij}]$ is a generalized Hadamard matrix of order $u\lambda$ with entries from $U$ which is a finite group of order $u$ (for short $\mathrm{G}\mathrm{H}(u,\lambda )$) such that whenever $i\mathrm{\ne }\mathrm{\ell }$ the set $\{d_{ij}{d}_{\mathrm{\ell }j}^{-1}\mathrm{\mid }1\le j\le u\lambda \}$ contains each element of $U$ exactly $\lambda$ times. In this paper, we construct $\mathrm{G}\mathrm{H}(q,q)$'s and $\mathrm{G}\mathrm{H}(q,q^2)$'s over additive groups of finite fields $\mathrm{G}\mathrm{F}(q)$'s by using some sorts of functions.