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 $\mathrm{ GH}(u,\,\lambda)$) such that whenever $i\neq \ell$ the set $\{ d_{ ij}d_{ \ell j}^{ -1}\,|\, 1\leq j\leq u\lambda \}$ contains each element of $U$ exactly $\lambda$ times. In this paper, we construct $\mathrm{ GH}(q,\,q)$'s and $\mathrm{ GH}(q,\,q^{ 2})$'s over additive groups of finite fields $\mathrm{ GF}(q)$'s by using some sorts of functions.
Categories: math.CO