Annihilator and twisted Euclidean duality for quasi-polycyclic codes
Abstract
Let f∈𝔽q[x] be a monic polynomial of degree m with f(0)≠ 0, and let ℛ=𝔽q[x]/⟨ f⟩. Under coefficient expansion, a quasi-polycyclic (QP) code of index n corresponds to an ℛ-submodule of ℛn. 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 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 𝔽q-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 ℛ and, when a suitable duality-preserving coordinate map exists, over 𝔽q. This gives binary and ternary stabilizer codes with minimum-distance lower bounds matching the best known bounds, most of which arise from rings ℛ that are not fields.
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BibTeX
@misc{bag2026annihilator,
title = {{Annihilator and twisted Euclidean duality for quasi-polycyclic codes}},
author = {Tushar Bag and Edgar Martínez-Moro and Daniel Panario},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26633},
eprint = {2609.26633},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26633},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Annihilator and twisted Euclidean duality for quasi-polycyclic codes
AU - Tushar Bag
AU - Edgar Martínez-Moro
AU - Daniel Panario
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26633
UR - https://arxiv.org/abs/2609.26633
AB - Let f∈𝔽_q[x] be a monic polynomial of degree m with f(0)≠ 0, and let ℛ=𝔽_q[x]/⟨ f⟩. Under coefficient expansion, a quasi-polycyclic (QP) code of index n corresponds to an ℛ-submodule of 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 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 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 ℛ and, when a suitable duality-preserving coordinate map exists, over 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 ℛ that are not fields.
ER -