EAQEC codes from polycyclic codes over the ring $R_l$
DOI:
https://doi.org/10.13069/jacodesmath.v13i2.390Keywords:
Semi-simple ring, Polycyclic codes, Hamming distances, Gray maps, Annihilator dual codesAbstract
In this paper, we study polycyclic codes over the ring \( R_l = \frac{\mathbb{F}_q[w]}{\langle w^l - 1 \rangle} \) with \( q = p^k \) where \( k \) is a positive integer and \( p \) is an odd prime. We explore LCD annihilator, self-dual, self-orthogonal codes over \( R_l \) and polycyclic codes over \( R_l \). Moreover, we provide a structure of entanglement-assisted quantum error-correcting codes based on the developed polycyclic codes through the inclusion of dual contained conditions. Subsequently, some LCD hull code-derived entanglement-assisted quantum error-correcting code examples are presented.
Accepted: 16 March 2026
Downloads
References
T. Abualrub and I. Siap, Reversible cyclic codes over Z4, Australas. J. Comb. 38 (2007) 195–206.
A. Alahmadi, S. Dougherty, A. Leroy and P. Solé, On the duality and the direction of polycyclic codes, Adv. Math. Commun. 10 (2016) 921–929.
A. Alahmadi, H. Islam, O. Prakash, P. Solé, A. Alkenanil, N. Muthana and R. Hijazi, New quantum codes from constacyclic codes over a non-chain ring, Quantum Information Processing 20 (2021) 60.
T. Bag and D. Panario, Quasi-polycyclic and skew quasi-polycyclic codes over Fq, Finite Fields and Their Applications 101 (2025) 102536.
E. R. Berlekamp, Algebraic coding theory (revised edition), World Scientific 2015.
I. F. Blake, Codes over certain rings, Information and Control 20 (1972) 396–404.
T. A. Brun, I. Devetak and H. Hsieh, Correcting quantum errors with entanglement, Science 314 (2006) 436–439.
A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory 44(4) (1998) 1369–1387.
Y. Cao, On constacyclic codes over finite chain rings, Finite Fields and Their Applications 24 (2013) 124–135.
B. Chen, Y. Fan, L. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields and Their Applications 18 (2012) 1217–1231.
R. Dastbasteh, F. Padashnick, P. M. Crespo, M. Grassl and J. Sharafi, Equivalence of constacyclic codes with shift constants of different orders, Designs Codes and Cryptography 93 (2025) 79–93.
X. Dong and S. Yin, The trace representation of λ-constacyclic codes over Fq, J. Liaoning Normal Univ. (Nat. Sci. ed.) 33 (2010) 129–131.
A. Fotue-Tabue, E. Martinez-Moro and J. T. Blackford, On polycyclic codes over a finite chain ring, Adv. Math. Commun. 14 (2020) 445–466.
R. Gokul, G. Karthick, M. Cruz, C. Durairajan and Giuliano G. La Guardia, Quantum Codes From the Polycyclic Codes over the Ring Fq[u, v]/<u2 = u, v2 = v, uv = vu>, J. Appl. Math. & Informatics 43 (2025) 1533–1548.
M. Güzeltepe and N. Aytaç, Quantum Codes from Codes over the Ring Rq, International Journal of Theoretical Physics 62 (2023) 26.
G. Karthick, Polycyclic codes over R, Communications in Combinatorics and Optimization 10 (2025) 371–379.
F. Li, Q. Yue and F. Liu, The weight distributions of constacyclic codes, Advances in Mathematics of Communications 11 (2017) 471–480.
F. Li and Q. Yue, The primitive idempotents and weight distributions of irreducible constacyclic codes, Designs, Codes and Cryptography 86 (2018) 771–784.
S. R. Lopez-Permouth, B. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun. 3 (2009) 227–234.
Om Prakash Pandey, Sachin Pathak, Awadhesh Kumar Shukla, Vipul Mishra, Ashish Kumar Upadhyay, A study of QECCs and EAQECCs construction from cyclic codes over the ring Fq + v1Fq + v2Fq + ··· + vsFq, Quantum Information Processing (2024) 23–31.
S. Patel, H. Islam and O. Prakash, (f, ρ, δ)-skew Polycyclic Codes and Their Applications to Quantum Codes, International Journal of Theoretical Physics 61 (2022) 47.
E. Prange. Cyclic Error-Correcting Codes in Two Symbols, Tech. rep. TN-57-103 1957.
W. Qi, On the polycyclic codes over Fq + uFq, Advances in Mathematics of Communications 18 (2024) 661–673.
W. Qi, The polycyclic codes over the finite field Fq, AIMS Mathematics 9 (2024) 29707–29717.
A. Singh, P. Sharma and O. Prakash, New quantum codes and (θ, δ, β)-cyclic codes, IEEE Access 12 (2024) 90345.
M. Shi, X. Li, Z. Sepasdar and P. Solé, Polycyclic codes as invariant subspaces, Finite Fields and Their Applications 68 (2020) 101760.
P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A 52 (1995) R2493.
Z. Tian, J. Gao and Y. Gao, Hulls of constacyclic codes over finite non-chain rings and their applications in quantum codes construction, Quantum Information Processing 23 (2024) 9.
S. Yadav, A. Singh and O. Prakash, Complementary dual skew polycyclic codes and their applications to EAQECCs, European Physical Journal Plus 138 (2023) 637.
B. Yildiz and N. Aydin, On cyclic codes over Z4 + uZ4 and their Z4-images, Int. J. Inf. Coding Theory 2 (2014) 226–237.
X. Zheng and B. Kong, Cyclic codes and λ1 + λ2u + λ3v + λ4uv-constacyclic codes over Fp + uFp + vFp + uvFp, Appl. Math. Comput. 306 (2017) 86–91.
