On the algebraic structure of quasi-polycyclic codes and new quantum codes

In this paper, we are interested in right (resp., left) quasi-polycyclic (QP) codes of length \(n=m\ell \) $n=m\ell$ with an associated vector \(a=(a_0,a_1,\ldots , a_{m-1})\in {\mathbb {F}}_{q}^{^m}\) $a=(a_0,a_1,\dots ,a_{m-1})\in {\mathbb{F}}_q^{{}^m}$ , which are a generalization of quasi-cyclic codes (QC) and quasi-twisted (QT) codes. They are defined as invariant subspaces of \({\mathbb {F}}_{_q}^{^{n}} \) ${\mathbb{F}}_{{}_q}^{{}^n}$ by the right (resp., left) QP operator \(\widetilde{T}_{\overrightarrow{a}}\) ${\tilde{T}}_{\overrightarrow{a}}$ (resp., \(\widetilde{T}_{\overleftarrow{a}}\) ${\tilde{T}}_{\overleftarrow{a}}$ ). A correspondence between the right (resp., left) \(\ell \) $\ell$ -QP codes and the linear codes of length \( \ell \) $\ell$ over the ring \(R_{{\overrightarrow{a},m}}:={\mathbb {F}}_{q}[x] / \left\langle x^m-\overrightarrow{a}(x)\right\rangle ,\ \overrightarrow{a}(x)=\sum _{i=0}^{m-1}a_ix^i\) $R_{\overrightarrow{a},m}:={\mathbb{F}}_q[x]/〈x^m,-,\overrightarrow{a},(x)〉,\overrightarrow{a}(x)={\sum }_{i=0}^{m-1}{a}_i{x}^i$ (resp., \( R_{{\overleftarrow{a},m}}:={\mathbb {F}}_{q}[x] /\left\langle x^m-\overleftarrow{a}(x)\right\rangle \) $R_{\overleftarrow{a},m}:={\mathbb{F}}_q[x]/〈x^m,-,\overleftarrow{a},(x)〉$ , \(\overleftarrow{a}(x)=\sum _{i=0}^{n-1}a_ix^{m-1-i}\) $\overleftarrow{a}(x)={\sum }_{i=0}^{n-1}{a}_i{x}^{m-1-i}$ ) is given. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials. Moreover, we also discuss the structure of the 1-generator QP codes, and we prove BCH-like and the Hartmann–Tzeng-like bounds on the minimum distance of QP codes. Finally, we give examples of new quantum codes derived from QP codes as an application of some of the results. Several of these codes are MDS.