On the structure of monomial codes and their generalizations

In this paper, we are interested in monomial codes with associated vector $a=(a_0, a_1,\ldots, a_{n-1}),$ introduced in \cite{Maria2017}, and more generally in linear codes invariant under a monomial matrix $M=\diag(a_0, a_1,\ldots, a_{n-1}) P_{\sigma}$ where $\sigma$ is a permutation and $P_{\sigma}$ its associated permutation matrix.We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix $M$. Next, we identify monomial codes with associated vector $a=(a_0,a_2,\ldots, a_{n-1})$ by the ideals of the polynomial ring $ R_{_{q,n}}:= \quot{{\Fq[x]}}{{\langle x^{n}-\prod_{i=0}^{n-1}a_i \rangle}},$ via a special isomorphism $\varphi_{_{\overline{a}}}$ which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of $\ell-$quasi-monomial ( $\ell-$QM) codes of length $n=m\ell,$ where on the one hand, we characterize them by the $ R_{_{q,m}}-$submodules of $ R_{_{q,m}}^{\ell}.$ On the other hand, $\ell-$QM codes are seen as additive monomial codes over the extension $\mathbb{F}_{q^{\ell}}/\Fq.$ So, as in the case of quasi-cyclic codes \cite{Guneri2018}, we characterize those codes that have $\mathbb{F}_{q^{\ell}}-$linear images with respect to a basis of the extension $ \mathbb{F}_{q^{\ell}}/\Fq,$ based on the CRT decomposition. Finally, we show that $\ell-$QM codes and additive monomial codes are asymptotically good.