$\theta$-Generalized monomial codes

Source abstract

In this paper we generalize cyclic codes to another more large linear codes, that is $\theta$-monomial codes. It is shown that for a $\theta$-monomial code, its Euclidean and $e$-Galois dual is also $\theta$-monomial code. Furthermore we present the equivalence between $\theta$-monomial codes and generalized monomial codes. By considering the skew polynomial ring, we show that $\theta$-monomial codes can relate to submodules under a condition and to ideals under other condition,this allow us to give a characterization of $\theta$-monomial codes. More results on the $e$-Galois dual of $\theta$-monomial codes are given with additional properties on self duality and self orthogonality. The Generalized $\theta$-monomial codes are discussed with their algebraic structure. The paper is closed by the investigation of the algebraic structure of $\theta$-monomial codes over the ring $\mathbb{F}_q+v\mathbb{F}_q$ where $v^2=v$. Received: 14 April 2024 | Accepted: 29 August 2024