Arithmetic properties of some families in $ \mathbb{F}_l[x]/\langle x^{p^sq^t}-1\rangle $ were obtained by using the cyclotomic classes of order 2 with respect to $ n = p^sq^t $, where $ p\equiv3\ \mathrm{mod}\ 4 $, $ \gcd(\phi(p^s), \phi(q^t)) = 2 $, $ l $ is a primitive root modulo $ q^t $, and $ \mathrm{ord}_{p^s}(l) = \phi(p^s)/2 $. The form of these cyclotomic classes enabled us to further generalize the results obtained in [
Citation: Juncheng Zhou, Hongfeng Wu. Cyclotomy, cyclotomic cosets and arithmetic properties of some families in $ \frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle} $[J]. AIMS Mathematics, 2026, 11(4): 10049-10078. doi: 10.3934/math.2026415
Arithmetic properties of some families in $ \mathbb{F}_l[x]/\langle x^{p^sq^t}-1\rangle $ were obtained by using the cyclotomic classes of order 2 with respect to $ n = p^sq^t $, where $ p\equiv3\ \mathrm{mod}\ 4 $, $ \gcd(\phi(p^s), \phi(q^t)) = 2 $, $ l $ is a primitive root modulo $ q^t $, and $ \mathrm{ord}_{p^s}(l) = \phi(p^s)/2 $. The form of these cyclotomic classes enabled us to further generalize the results obtained in [
| [1] |
K. Kumar, S. Betra, Cyclotomy, cyclotomic cosets and arithmetic properties of some families in $\frac{\mathbb{F}_l[x]}{\langle x^{p^tq}-1\rangle}$, Asian-Eur. J. Math., 2 (2023), 2350023. https://doi.org/10.1142/S1793557123500237 doi: 10.1142/S1793557123500237
|
| [2] |
C. Ding, T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140–166. https://doi.org/10.1006/ffta.1998.0207 doi: 10.1006/ffta.1998.0207
|
| [3] |
F. Li, Q. Yue, The primitive idempotents and weight distributions of irreducible constacyclix codes, Des. Codes Cryptogr., 86 (2018), 771–784. https://doi.org/10.1007/s10623-017-0356-2 doi: 10.1007/s10623-017-0356-2
|
| [4] |
C. Ding, Cyclotomic constructions of cyclic codes with length being the product of two primes, IEEE Trans. Inform. Theory, 58 (2012), 2231–2236. https://doi.org/10.1109/TIT.2011.2176915 doi: 10.1109/TIT.2011.2176915
|
| [5] | R. Singh, M. Pruthi, Primitive idempotents of irreducible quadratic residue cyclic codes of length $p^nq^m$, Int. J. Algrbra, 5 (2011), 285–294. |
| [6] |
F. Li, Q. Yue, The minimum Hamming distances of irreducible cyclic codes, Finite Fields Appl., 29 (2014), 225–242. https://doi.org/10.1016/j.ffa.2014.05.003 doi: 10.1016/j.ffa.2014.05.003
|
| [7] |
Z. Shi, F. Fu, The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes, Cryptogr. Commun., 12 (2020), 29–52. https://doi.org/10.1007/s12095-019-00362-w doi: 10.1007/s12095-019-00362-w
|
| [8] |
G. K. Bakshi, M. Raka, Minimal cyclic codes of length $p^nq$, Finite Fields Appl., 9 (2003), 432–448. https://doi.org/10.1016/S1071-5797(03)00023-6 doi: 10.1016/S1071-5797(03)00023-6
|
| [9] |
A. Sahni, P. T. Sehgal, Minimal cyclic codes of length $p^nq$, Finite Fields Appl., 18 (2012), 1017–1036. https://doi.org/10.1016/j.ffa.2012.07.003 doi: 10.1016/j.ffa.2012.07.003
|
| [10] |
S. Jain, S. Betra, Cyclotomy and arithmetic properties of some families in $\mathbb{Z}[x]/\langle x^pq-1\rangle$, Asian-Eur. J. Math., 13 (2020), 2050077. https://doi.org/10.1142/S1793557120500771 doi: 10.1142/S1793557120500771
|
| [11] |
S. Jain, S. Batra, Cyclic self dual codes of length $pq$ over $\mathbb{Z}_4$, J. Discr. Math. Sci. Cryptogr., 15 (2022), 2305–2319. https://doi.org/10.1080/09720529.2020.1823680 doi: 10.1080/09720529.2020.1823680
|
| [12] | F. J. MacWilliams, N. J. Z. Sloane, Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. |
| [13] | D. M. Burton, Elementary Number Theory, New Delhi: Tate McGra-Hill Publishers, 2007. |
| [14] | G. Hardy, E. Wright, An Introduction to the Theory of Numbers, 5Eds., Oxford: Clarendon Press, 1984. |
Juncheng Zhou, Hongfeng Wu. Cyclotomy, cyclotomic cosets and arithmetic properties of some families in $ \frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle} $[J]. AIMS Mathematics, 2026, 11(4): 10049-10078. doi: 10.3934/math.2026415
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