In this article, we study polycyclic codes over the ring R=Fq[v]/⟨v2−1⟩, where q=pm with p being an odd prime. First, we introduce polycyclic codes and sequential codes over R, and characterize the structural properties of these polycyclic codes. Next, we analyze the Euclidean dual codes, annihilator dual codes, annihilator self-orthogonal codes, and annihilator linear complementary dual (LCD) codes associated with this family of codes. Finally, some asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) are constructed from polycyclic codes over R. Moreover, the parameters of our AEAQECCs are new in the existing literature.In this article, we study polycyclic codes over the ring R=Fq[v]/⟨v2−1⟩, where q=pm with p being an odd prime. First, we introduce polycyclic codes and sequential codes over R, and characterize the structural properties of these polycyclic codes. Next, we analyze the Euclidean dual codes, annihilator dual codes, annihilator self-orthogonal codes, and annihilator linear complementary dual (LCD) codes associated with this family of codes. Finally, some asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) are constructed from polycyclic codes over R. Moreover, the parameters of our AEAQECCs are new in the existing literature.

AxiomsFeb 2, 20260 citations

On construction of quantum codes with dual-containing quasi-cyclic codes

Chaofeng Guan, Ruihu Li, Liangdong Lü +2 more

arXiv:2210.08716Quantum Information ProcessingJul 26, 202213 citations

Annihilator and twisted Euclidean duality for quasi-polycyclic codes

Tushar Bag, Edgar Martı́nez-Moro, Daniel Panario

Let $f\in\mathbb F_q[x]$ be a monic polynomial of degree $m$ with $f(0)\ne 0$ , and let $\mathcal R=\mathbb F_q[x]/\langle f\rangle$ . Under coefficient expansion, a quasi-polycyclic (QP) code of index $n$ corresponds to an $\mathcal R$ -submodule of $\mathcal R^n$ . In this paper, we study QP codes with respect to the annihilator duality. We show that this form is non-degenerate and that the annihilator dual of a QP code is again a QP code. We also give an equivalent description of the dual in terms of the $\mathcal R$ -valued dot product, which leads to self-orthogonality criteria. We determine the Gram matrix of the annihilator form and obtain an explicit formula for its determinant. In coefficient coordinates, this shows that the annihilator dual can be viewed as a twisted Euclidean dual. Using this description, we characterize when a coordinatewise $\mathbb F_q$ -linear map converts annihilator duality into ordinary Euclidean duality. For squarefree $f$ , we show that annihilator duality decomposes into ordinary Euclidean duality on the components arising from the Chinese Remainder Theorem. This gives simple criteria for self-orthogonal, self-dual, dual-containing, and complementary-dual QP codes. We show how the annihilator dual interacts with the Hamming weight enumerator and compute the MacWilliams transform associated with that duality. Finally, we apply these results to Calderbank--Shor--Steane and Steane-enlarged quantum-code constructions over $\mathcal R$ and, when a suitable duality-preserving coordinate map exists, over $\mathbb F_q$ . This gives binary and ternary stabilizer codes with minimum-distance lower bounds matching the best known bounds, most of which arise from rings $\mathcal R$ that are not fields.Let $f\in\mathbb F_q[x]$ be a monic polynomial of degree $m$ with $f(0)\ne 0$, and let $\mathcal R=\mathbb F_q[x]/\langle f\rangle$. Under coefficient expansion, a quasi-polycyclic (QP) code of index $n$ corresponds to an $\mathcal R$-submodule of $\mathcal R^n$. In this paper, we study QP codes with respect to the annihilator duality. We show that this form is non-degenerate and that the annihilator dual of a QP code is again a QP code. We also give an equivalent description of the dual in terms of the $\mathcal R$-valued dot product, which leads to self-orthogonality criteria. We determine the Gram matrix of the annihilator form and obtain an explicit formula for its determinant. In coefficient coordinates, this shows that the annihilator dual can be viewed as a twisted Euclidean dual. Using this description, we characterize when a coordinatewise $\mathbb F_q$-linear map converts annihilator duality into ordinary Euclidean duality. For squarefree $f$, we show that annihilator duality decomposes into ordinary Euclidean duality on the components arising from the Chinese Remainder Theorem. This gives simple criteria for self-orthogonal, self-dual, dual-containing, and complementary-dual QP codes. We show how the annihilator dual interacts with the Hamming weight enumerator and compute the MacWilliams transform associated with that duality. Finally, we apply these results to Calderbank--Shor--Steane and Steane-enlarged quantum-code constructions over $\mathcal R$ and, when a suitable duality-preserving coordinate map exists, over $\mathbb F_q$. This gives binary and ternary stabilizer codes with minimum-distance lower bounds matching the best known bounds, most of which arise from rings $\mathcal R$ that are not fields.

arXiv:2609.26633Sep 22, 20260 citations

On the duals of quasi-cyclic codes and their application to quantum codes

Shivanshu Benjwal, Maheshanand Bhaintwal

Quantum Information ProcessingMar 23, 20243 citations

Quasi-polycyclic and skew quasi-polycyclic codes over Fq

Tushar Bag, Daniel Panario

Finite Fields and Their ApplicationsNov 1, 20243 citations

Let

f\in\mathbb F_q[x]

be a monic polynomial of degree

m

with

f(0)\ne 0

, and let

\mathcal R=\mathbb F_q[x]/\langle f\rangle

. Under coefficient expansion, a quasi-polycyclic (QP) code of index

n

corresponds to an

\mathcal R

-submodule of

\mathcal R^n

. In this paper, we study QP codes with respect to the annihilator duality. We show that this form is non-degenerate and that the annihilator dual of a QP code is again a QP code. We also give an equivalent description of the dual in terms of the

\mathcal R

-valued dot product, which leads to self-orthogonality criteria. We determine the Gram matrix of the annihilator form and obtain an explicit formula for its determinant. In coefficient coordinates, this shows that the annihilator dual can be viewed as a twisted Euclidean dual. Using this description, we characterize when a coordinatewise

\mathbb F_q

-linear map converts annihilator duality into ordinary Euclidean duality. For squarefree

f

, we show that annihilator duality decomposes into ordinary Euclidean duality on the components arising from the Chinese Remainder Theorem. This gives simple criteria for self-orthogonal, self-dual, dual-containing, and complementary-dual QP codes. We show how the annihilator dual interacts with the Hamming weight enumerator and compute the MacWilliams transform associated with that duality. Finally, we apply these results to Calderbank--Shor--Steane and Steane-enlarged quantum-code constructions over

\mathcal R

and, when a suitable duality-preserving coordinate map exists, over

\mathbb F_q

. This gives binary and ternary stabilizer codes with minimum-distance lower bounds matching the best known bounds, most of which arise from rings

\mathcal R

that are not fields.