Let $ p $ be a prime and $ m $ a positive integer. This paper investigates cyclic and self-dual codes of length $ n $ over the local Frobenius non-chain ring $ R = \mathrm{GR}(p^3, m)[v] $, with $ v^2 = p^2 \alpha $, $ \alpha \in \mathbb{F}^*_{p^m} $, and $ p v = 0 $. First, we characterize the algebraic structure of cyclic codes of arbitrary length over $ R $. When $ \gcd(n, p) = 1 $, explicit generator polynomials are determined, and the corresponding dual and self-orthogonal structures are derived. A key result of this study is the proof that self-dual cyclic codes do not exist over $ R $. In addition, the enumeration formula for cyclic LCD codes is given by $ 2^{e_1+\frac{e_2}{2}}. $ Several examples and tables are provided to illustrate the theoretical findings and the derived mass formulas for cyclic self-orthogonal and LCD codes.
Citation: Sami Alabiad, Alhanouf Ali Alhomaidhi. Various structures of cyclic codes and LCD codes over $ GR(p^3, m)[v]/\langle v^2-p^2\alpha, p v\rangle $[J]. AIMS Mathematics, 2025, 10(11): 27535-27559. doi: 10.3934/math.20251211
Let $ p $ be a prime and $ m $ a positive integer. This paper investigates cyclic and self-dual codes of length $ n $ over the local Frobenius non-chain ring $ R = \mathrm{GR}(p^3, m)[v] $, with $ v^2 = p^2 \alpha $, $ \alpha \in \mathbb{F}^*_{p^m} $, and $ p v = 0 $. First, we characterize the algebraic structure of cyclic codes of arbitrary length over $ R $. When $ \gcd(n, p) = 1 $, explicit generator polynomials are determined, and the corresponding dual and self-orthogonal structures are derived. A key result of this study is the proof that self-dual cyclic codes do not exist over $ R $. In addition, the enumeration formula for cyclic LCD codes is given by $ 2^{e_1+\frac{e_2}{2}}. $ Several examples and tables are provided to illustrate the theoretical findings and the derived mass formulas for cyclic self-orthogonal and LCD codes.
| [1] |
I. F. Blake, Codes over certain rings, Inf. Contr., 20 (1972), 396–404. http://doi.org/10.1016/S0019-9958(72)90223-9 doi: 10.1016/S0019-9958(72)90223-9
|
| [2] |
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inf. Theory, 40 (1994), 301–319. http://doi.org/10.1109/18.312154 doi: 10.1109/18.312154
|
| [3] |
A. R. Calderbank, N. J. A. Sloane, Modular and $p$-adic codes, Des. Codes Cryptogr., 6 (1995), 21–35. http://doi.org/10.1007/BF01390768 doi: 10.1007/BF01390768
|
| [4] |
P. Kanwar, S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334–352. http://doi.org/10.1006/ffta.1997.0189 doi: 10.1006/ffta.1997.0189
|
| [5] |
A. Bonnecaze, P. Udaya, Cyclic codes and self-dual codes over $\mathbb{F}_2 + u\mathbb{F}_2$, IEEE Trans. Inf. Theory, 45 (1999), 1250–1255. http://doi.org/10.1109/18.761278 doi: 10.1109/18.761278
|
| [6] |
H. Q. Dinh, S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inf. Theory, 50 (2004), 1728–1744. http://doi.org/10.1109/TIT.2004.831789 doi: 10.1109/TIT.2004.831789
|
| [7] |
B. Yildiz, S. Karadeniz, Cyclic codes over $\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2$, Des. Codes Cryptogr., 58 (2011), 221–234. http://doi.org/10.1007/s10623-010-9399-3 doi: 10.1007/s10623-010-9399-3
|
| [8] |
A. K. Singh, P. K. Kewat, On cyclic codes over the ring $\mathbb{Z}_p[u]/\langle u^k \rangle$, Des. Codes Cryptogr., 74 (2015), 1–13. http://doi.org/10.1007/s10623-013-9843-2 doi: 10.1007/s10623-013-9843-2
|
| [9] |
B. Kim, Y. Lee, Classification of self-dual cyclic codes over the chain ring $\mathbb{Z}_p[u]/\langle u^3 \rangle$, Des. Codes Cryptogr., 88 (2020), 2247–2273. http://doi.org/10.1007/s10623-020-00776-1 doi: 10.1007/s10623-020-00776-1
|
| [10] | E. M. Rains, N. J. A. Sloane, Self-dual codes, in V. S. Pless, W. C. Huffman (Eds.), Handbook of Coding Theory, Amsterdam: North-Holland, 1998. https://doi.org/10.48550/arXiv.math/0208001 |
| [11] |
T. A. Gulliver, V. K. Bhargava, Some best rate $\frac{1}{p}$ and rate $\frac{p-1}{p}$ systematic quasi-cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 552–555. http://doi.org/10.1109/18.79911 doi: 10.1109/18.79911
|
| [12] |
M. Shi, D. Huang, L. Sok, P. Solé, Double circulant self-dual and LCD codes over Galois rings, Adv. Math. Commun., 13 (2019), 171–183. http://doi.org/10.3934/amc.2019011 doi: 10.3934/amc.2019011
|
| [13] |
M. Shi, H. Zhu, L. Sok, P. Solé, On self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q + u\mathbb{F}_q$, Cryptogr. Commun., 12 (2020), 53–70. https://doi.org/10.1007/s12095-019-00363-9 doi: 10.1007/s12095-019-00363-9
|
| [14] |
J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337–342. http://doi.org/10.1016/0012-365X(92)90563-U doi: 10.1016/0012-365X(92)90563-U
|
| [15] |
M. Shi, Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields Appl., 39 (2016), 159–178. https://doi.org/10.1016/j.ffa.2016.01.010 doi: 10.1016/j.ffa.2016.01.010
|
| [16] |
M. Shi, H. Zhu, L. Qian, P. Solé, On self-dual four circulant codes, Int. J. Found. Comput. Sci., 29 (2018), 1143–1150. https://doi.org/10.1142/S0129054118500259 doi: 10.1142/S0129054118500259
|
| [17] |
M. Alshuhail, A. Betty, A. Galvez, Duality of codes over non-unital rings of order six, AIMS Math., 10 (2025), 4934–4951. http://doi.org/10.3934/math.2025839 doi: 10.3934/math.2025839
|
| [18] |
D. Xie, S. Zhu, Two families of self-orthogonal codes with applications in LCD codes and optimally extendable codes, Finite Fields Appl., 108 (2025), 102659. https://doi.org/10.1016/j.ffa.2025.102659 doi: 10.1016/j.ffa.2025.102659
|
| [19] |
H. Islam, O. Prakash, Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring, Cryptogr. Commun., 14 (2022), 59–73. https://doi.org/10.1007/s12095-021-00516-9 doi: 10.1007/s12095-021-00516-9
|
| [20] |
S. Li, M. Shi, H. Liu, Several constructions of optimal LCD codes over small finite fields, Cryptogr. Commun., 16 (2024), 779–800. https://doi.org/10.1007/s12095-024-00699 doi: 10.1007/s12095-024-00699
|
| [21] |
L. Sok, M. Shi, P. Solé, Construction of optimal LCD codes over large finite fields, Finite Fields Appl., 50 (2018), 138–153. https://doi.org/10.1016/j.ffa.2017.11.007 doi: 10.1016/j.ffa.2017.11.007
|
| [22] |
N. Aydin, Y. Cengellenmis, A. Dertli, On some constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2 - 1\rangle$, their $\mathbb{Z}_4$ images, and new codes, Des. Codes Cryptogr., 86 (2018), 1249–1255. http://doi.org/10.1007/s10623-017-0392-y doi: 10.1007/s10623-017-0392-y
|
| [23] |
M. Shi, F. Özbudak, L. Xu, P. Solé, LCD codes from tridiagonal Toeplitz matrices, Finite Fields Appl., 75 (2021), 101892. http://doi.org/10.1016/j.ffa.2021.101892 doi: 10.1016/j.ffa.2021.101892
|
| [24] |
C. A. Castillo-Guillén, C. Rentería-Márquez, H. Tapia-Recillas, Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, Finite Fields Appl., 43 (2017), 1–21. https://doi.org/10.1016/j.ffa.2016.08.004 doi: 10.1016/j.ffa.2016.08.004
|
| [25] |
H. S. Choi, B. Kim, Various structures of cyclic codes over the non-Frobenius ring $\mathbb{F}_p[u, v]/\langle u^2, v^2, uv, vu \rangle$, Adv. Math. Commun., 18 (2024), 304–327. http://doi.org/10.3934/amc.2023030 doi: 10.3934/amc.2023030
|
| [26] |
S. T. Dougherty, E. Saltürk, S. Szabo, On codes over Frobenius rings: Generating characters, MacWilliams identities and generator matrices, Appl. Algebra Eng. Commun. Comput., 30 (2019), 193–206. https://doi.org/10.1007/s00200-019-00384-0 doi: 10.1007/s00200-019-00384-0
|
| [27] |
S. H. Saif, Constructions and enumerations of self-dual and LCD double circulant codes over a local ring, Mathematics, 13 (2025), 3527. https://doi.org/10.3390/math13213527 doi: 10.3390/math13213527
|
| [28] |
S. H. Saif, Structures, ranks and minimal distances of cyclic codes over $\mathbb{Z}_{p^2} + u\mathbb{Z}_{p^2}$, Mathematics, 13 (2025), 3354. https://doi.org/10.3390/math13203354 doi: 10.3390/math13203354
|
| [29] |
S. Alabiad, A. A. Alhomaidhi, N. A. Alsarori, On linear codes over local rings of order $p^4$, Mathematics, 12 (2024), 3069. http://doi.org/10.3390/math12193069 doi: 10.3390/math12193069
|
| [30] |
S. Alabiad, A. A. Alhomaidhi, Cyclic, LCD, and self-dual codes over the non-Frobenius ring $\mathrm{GR}(p^2, m)[u]/\langle u^2, pu\rangle$, Mathematics, 13 (2025), 3193. https://doi.org/10.3390/math13193193 doi: 10.3390/math13193193
|
Figures(2) / Tables(4)
Sami Alabiad, Alhanouf Ali Alhomaidhi. Various structures of cyclic codes and LCD codes over $ GR(p^3, m)[v]/\langle v^2-p^2\alpha, p v\rangle $[J]. AIMS Mathematics, 2025, 10(11): 27535-27559. doi: 10.3934/math.20251211
DownLoad: