Codes of the Reed-Muller type on a finite abelian group (Q1276780)

Let \(G\) be a finite abelian group of exponent \(q\) and denote by \(T_q\) the group of complex \(q\)-th roots of unity. Let \(T_{q}^{G}\) denote the set of functions on \(G\) with values in \(T_q\). The group \(\widehat{G}\) of characters of \(G\) is isomorphic to \(G\), and elements \(\chi_\alpha \in \widehat{G}\) can be numbered by elements \(\alpha\in G\). In the paper, a first order code \(RM_{1}(G)\) of the Reed-Muller type on a group \(G\) is defined as \[ RM_{1}(G)=\{f\in T_{q}^{G} \mid \exists {_{\alpha,\sigma\in G}} \forall {_{x\in G}} f(x)=\chi{_\alpha}(\sigma x)\}. \] Introducing the derivative of a function \(f\in T{^G}\) as \[ {df \over d\sigma}(x) = \overline{f(x)}f(\sigma x), \;\forall {_{x\in G}}, \] the paper defines the \(r\)-th (\(r=2,3,\ldots\)) order codes \(RM{_r}(G)\) of Reed-Muller type as \[ RM{_r}(G) = \Biggl\{ f\in T_{q}^{G} \mid \forall{_{\sigma\in G}} {df\over d\sigma}\in RM{_{r-1}}(G)\Biggr\}. \] In the paper, several algebraic properties of the codes \(RM{_r}(G)\), which are well known for standard Reed-Muller codes, are proved.