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http://dx.doi.org/10.4134/JKMS.j210357

CONSTRUCTION FOR SELF-ORTHOGONAL CODES OVER A CERTAIN NON-CHAIN FROBENIUS RING  

Kim, Boran (Department of Mathematics Education Kyungpook National University)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.1, 2022 , pp. 193-204 More about this Journal
Abstract
We present construction methods for free self-orthogonal (self-dual or Type II) codes over ℤ4[v]/〈v2 + 2v〉 which is one of the finite commutative local non-chain Frobenius rings of order 16. By considering their Gray images on ℤ4, we give a construct method for a code over ℤ4. We have some new and optimal codes over ℤ4 with respect to the minimum Lee weight or minimum Euclidean weight.
Keywords
Frobenius ring; non-chain ring; self-orthogonal code; code over ${\mathbb{Z}}_4$; optimal code;
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1 B. Kim and Y. Lee, Lee weights of cyclic self-dual codes over Galois rings of characteristic p2, Finite Fields Appl. 45 (2017), 107-130. https://doi.org/10.1016/j.ffa.2016.11.015   DOI
2 B. Kim and Y. Lee, A mass formula for cyclic codes over Galois rings of characteristic p3, Finite Fields Appl. 52 (2018), 214-242. https://doi.org/10.1016/j.ffa.2018.04.005   DOI
3 V. S. Pless and W. C. Huffman, Handbook of coding theory, Volume I, Elsevier, North Holland, 1998.
4 M. Shi, H. Zhu, L. Qian, L. Sok, and P. Sole, On self-dual and LCD double circulant and double negacirculant codes over 𝔽q + u𝔽q, Cryptogr. Commun. 12 (2020), no. 1, 53-70. https://doi.org/10.1007/s12095-019-00363-9   DOI
5 S. Ling and P. Sole, Type II codes over &3x1D405;4 +u&3x1D405;4, European J. Combin. 22 (2001), no. 7, 983-997. https://doi.org/10.1006/eujc.2001.0509   DOI
6 M. Shi, S. Zhu, and S. Yang, A class of optimal p-ary codes from one-weight codes over 𝔽p[u]〈hum〉, J. Franklin Inst. 350 (2013), no. 5, 929-937. https://doi.org/10.1016/j.jfranklin.2012.05.014   DOI
7 A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett. 78 (1997), no. 3, 405-408. https://doi.org/10.1103/PhysRevLett.78.405   DOI
8 A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory 44 (1998), no. 4, 1369-1387. https://doi.org/10.1109/18.681315   DOI
9 Y. Cao and Y. Cao, Negacyclic codes over the local ring ℤ4[v]]〈hv2 + 2v]〉 of oddly even length and their Gray images, Finite Fields Appl. 52 (2018), 67-93. https://doi.org/10.1016/j.ffa.2018.03.005   DOI
10 Y. Choie and N. Kim, The complete weight enumerator of type II codes over ℤ2m and Jacobi forms, IEEE Trans. Inform. Theory 47 (2001), no. 1, 396-399. https://doi.org/10.1109/18.904543   DOI
11 B. Kim, C. Kim, S. Kwon, and Y. Kwon, Jacobi forms over number fields from linear codes, submitted.
12 N. Han, B. Kim, B. Kim, and Y. Lee, Infinite families of MDR cyclic codes over ℤ4 via constacyclic codes over ℤ4[u]〈hu2 - 1〉, Discrete Math. 343 (2020), no. 3, 111771, 12 pp. https://doi.org/10.1016/j.disc.2019.111771   DOI
13 E. Bannai, S. T. Dougherty, M. Harada, and M. Oura, Type II codes, even unimodular lattices, and invariant rings, IEEE Trans. Inform. Theory 45 (1999), no. 4, 1194-1205. https://doi.org/10.1109/18.761269   DOI
14 Y. Cao and Y. Cao, Complete classification for simple root cyclic codes over the local ring ℤ4[v]]〈hv2 + 2v]〉, Cryptogr. Commun. 12 (2020), no. 2, 301-319. https://doi.org/10.1007/s12095-019-00403-4   DOI
15 Database of ℤ4 codes [online], http://Z4Codes.info.
16 B. Kim, Y. Lee, and J. Doo, Classification of cyclic codes over a non-Galois chain ring ℤsub>p[u]〈hu3〉, Finite Fields Appl. 59 (2019), 208-237. https://doi.org/10.1016/j.ffa.2019.06.003   DOI
17 M. Shi, L. Qian, L. Sok, N. Aydin, and P. Sole, On constacyclic codes over ℤ4[u]〈hu2-1〉 and their Gray images, Finite Fields Appl. 45 (2017), 86-95. https://doi.org/10.1016/j.ffa.2016.11.016   DOI
18 B. Yildiz and S. Karadeniz, Linear codes over ℤ4 + uℤ4: MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl. 27 (2014), 24-40. https://doi.org/10.1016/j.ffa.2013.12.007   DOI
19 S. T. Dougherty, E. Salturk, and S. Szabo, On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices, Appl. Algebra Engrg. Comm. Comput. 30 (2019), no. 3, 193-206. https://doi.org/10.1007/s00200-019-00384-0   DOI
20 J. Y. Hyun, B. Kim, and M. Na, Construction of minimal linear codes from multivariable functions, Adv. Math. Commun. 15 (2021), no. 2, 227-240. https://doi.org/10.3934/amc.2020055   DOI
21 S. Karadeniz, S. T. Dougherty, and B. Yildiz, Constructing formally self-dual codes over Rk, Discrete Appl. Math. 167 (2014), 188-196. https://doi.org/10.1016/j.dam.2013.11.017   DOI