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 $\mathrm{GH}(u,\,\lambda)$) 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

Some classical views on the parameters of the Grothendieck-Teichmüeller group

We present two new formulas concerning behaviors of the standard parameters of the Grothendieck-Teichmüller group GT , and discuss their relationships with classical mathematics. First, considering a non-Galois etale cover of P1 {0 1 infinity} of degree 4, we present a newtype equation satisfied by the Galois image in GT . Second, a certain equation in GL 2 (Z||Z2 ) satisfied by every element of GT is derived as an application of (profinite) free differential calculus.</p