On additive cyclic codes over $\mathbb{F}_4+u\mathbb{F}_4$
DOI:
https://doi.org/10.13069/jacodesmath.v12i3.356Keywords:
Additive code, Cyclic codes, Quasi cyclic code, Symplectic dual, Gray mapAbstract
This article studies additive cyclic codes over $R = \mathbb{F}_4+u\mathbb{F}_4$, where $u^2 = 0$. We obtain generator polynomials for these codes and provide necessary and sufficient conditions for additive codes to be self-orthogonal and self-dual codes over $R$ with respect to the symplectic inner product. Additive self-orthogonal codes over $\mathbb{F}_4$ with respect to the symplectic inner product are used to construct quantum codes. We demonstrate that the Gray image of additive self-orthogonal codes over $R$ results in additive self-orthogonal codes over $\mathbb{F}_4$. Additionally, we prove that binary self-orthogonal codes can be obtained from additive self-orthogonal codes over $R$ with respect to the symplectic inner product.
Accepted: 20 February 2025
Downloads
References
T. Abualrub, N. Aydin, I. Aydogdu, Optimal binary codes derived from F2F4-additive cyclic codes, J. Appl. Math. Comput. 64(1–2) (2020) 71–87.
T. Abualrub, I. Siap, N. Aydin, Z2Z4-additive cyclic codes, IEEE Trans. Inform. Theory 60(3) (2014) 1508–1514.
A. Agrawal, G. K. Verma, R. K. Sharma, Galois LCD codes over Fq + uFq + vFq + uvFq, Bulletin of the Australian Mathematical Society 107(2) (2022) 330–341.
I. Aydogdu, T. Abualrub, I. Siap, On Z2Z2[u]-additive codes, Int. J. Comput. Math. 92(9) (2015) 1806–1814.
A. R. Calderbank, E. M. Rains, P. W. Shor, N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory 44(4) (1998) 1369–1387.
P. Delsarte, V. I. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory 44(6) (1998) 2477–2504.
L. Diao, J. Gao, J. Lu, Some results on ZpZp[v]-additive cyclic codes, Adv. Math. Commun. 14(4) (2020) 555–572.
C. Ding, Cyclic codes over finite fields, Designs from Linear Codes (2018) 89–109.
P. Gaborit, V. Pless, P. Solé, O. Atkin, Type II codes over F4, Finite Fields Appl. 8(2) (2002) 171–183.
H. Islam, E. Martinez-Moro, O. Prakash, Cyclic codes over a non-chain ring Re,q and their application to LCD codes, Discret. Math. 344 (2021) 112545.
H. Islam, O. Prakash, A study of cyclic and constacyclic codes over Z4 + uZ4 + vZ4, Int. J. Inf. Coding Theory 5 (2018) 155–168.
A. Ketkar, A. Klappenecker, S. Kumar, P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory 52(11) (2006) 4892–4914.
C. Li, C. Ding, S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inform. Theory 63 (2016) 4344–4356.
S. Ling, P. Solé, Type II codes over F4 + uF4, Eur. J. Comb. 22 (2001) 983–997.
E. Martinez-Moro, K. Otal, F. Ozbudak, Additive cyclic codes over finite commutative chain rings, Discrete Math. 341(7) (2018) 1873–1884.
M. Shi, S. Chu, J.-L. Kim, Classification of type I codes over F4 + uF4, Journal of Applied Mathematics and Computing 69 (2023) 3021–3037.
M. Shi, R. Wu, P. Solé, Asymptotically good additive cyclic codes exist, IEEE Communications Letters 22(10) (2018) 1980–1983.
B. Srinivasulu, M. Bhaintwal, Reversible cyclic codes over F4 + uF4 and their applications to DNA codes, 7th International Conference on Information Technology and Electrical Engineering (ICITEE) (2015) 101–105.
B. Srinivasulu, M. Bhaintwal, Z2(Z2 + uZ2)-additive cyclic codes and their duals, Discret. Math. Algorithms Appl. 8(2) (2016) 1650027.
M. Sucheta Dutt, R. Sehmi, On cyclic codes over finite chain rings, Journal of Physics: Conference Series 1850 (2021).
T. Yao, S. Zhu, ZpZps-additive cyclic codes are asymptotically good, Cryptogr. Commun. 12(2) (2020) 253–264.
T. Yao, S. Zhu, X. Kai, Asymptotically good ZprZps-additive cyclic codes, Finite Fields Appl. 63 (2020) 101633.
