doi: 10.3934/amc.2021045
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On BCH split metacyclic codes

Normandie Univ., UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme, 14000 Caen, France

Received  April 2021 Revised  August 2021 Early access September 2021

Recently, Borello and Jamous have investigated some lower bounds on the dimension and minimum distance for dihedral codes, in analogy with the theory of BCH codes. In this paper, we extend some of their results to split metacyclic codes, that is, codes over semidirect products of cyclic groups.

Citation: Angelot Behajaina. On BCH split metacyclic codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021045
References:
[1]

S. Ball, A Course in Algebraic Error-Correcting Codes, Compact Textbooks in Mathematics, 2020. doi: 10.1007/978-3-030-41153-4.  Google Scholar

[2]

M. BarbierC. Chabot and G. Quintin, On quasi-cyclic codes as a generalization of cyclic codes, Finite Fields Appl., 18 (2012), 904-919.  doi: 10.1016/j.ffa.2012.06.003.  Google Scholar

[3]

L. M. J. Bazzi and S. K. Mitter, Some randomized code constructions from group actions, IEEE Trans. Inform. Theory, 52 (2006), 3210-3219.  doi: 10.1109/TIT.2006.876244.  Google Scholar

[4]

S. D. Berman, On the theory of group codes, Cybernetics, 3 (1969), 25-31.  doi: 10.1007/BF01072842.  Google Scholar

[5]

J. J. BernalA. del Rio and J. J. Simon, An intrinsical description of group codes, Designs, Codes and Cryptography, 51 (2009), 289-300.  doi: 10.1007/s10623-008-9261-z.  Google Scholar

[6]

F. BernhardtP. Landrock and O. Manz, The extended Golay codes considered as ideals, J. Comb. Theory, Series A, 55 (1990), 235-246.  doi: 10.1016/0097-3165(90)90069-9.  Google Scholar

[7]

M. Borello, J. de la Cruz and W. Willems, On checkable codes in group algebras, Journal of Algebra and its Applications, (2021). doi: 10.1142/S0219498822501250.  Google Scholar

[8]

M. Borello and A. Jamous, Dihedral codes with prescribed minimum distance, In: Bajard J. C., Topuzoglu A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science, col 12545. Springer, Cham. (2021). doi: 10.1007/978-3-030-68869-1_8.  Google Scholar

[9]

M. BorelloP. Moree and P. Solé, Asymptotic performance of metacyclic codes, Discrete Math., 343 (2020), 111885.  doi: 10.1016/j.disc.2020.111885.  Google Scholar

[10]

M. Borello and W. Willems, Group codes over fields are asymptotically good, Finite Fields Appl., 68 (2020), 101738.  doi: 10.1016/j.ffa.2020.101738.  Google Scholar

[11]

R. C. Bose and D. K. Ray-Chaudhuri, On a class of error correcting binary group codes, Information and control, 3 (1960), 68-79.  doi: 10.1016/S0019-9958(60)90287-4.  Google Scholar

[12]

P. Charpin, Une généralisation de la construction de Berman des codes de Reed-Muller p-aire, Comm. Algebra, 16 (1988), 2231-2246.  doi: 10.1080/00927878808823689.  Google Scholar

[13]

C. L. ChenW. W. Peterson and E. J. Weldon Jr., Some results on quasi-cyclic codes, Information and control, 15 (1969), 407-423.  doi: 10.1016/S0019-9958(69)90497-5.  Google Scholar

[14]

J. H. ConwayS. J. Lomonaco Jr. and N. J. A. Sloane, A [45, 13] code with minimal distance 16, Discrete Math., 83 (1990), 213-217.  doi: 10.1016/0012-365X(90)90007-5.  Google Scholar

[15]

M. Grassl, Codetables, http://www.codetables.de/. Google Scholar

[16]

A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.   Google Scholar

[17] W. C. HuffmanJ.-L. Kim and P. Solé, Concise Encyclopedia of Coding Theory, CRC Press, 2021.  doi: 10.1201/9781315147901.  Google Scholar
[18]

S. Jitman, S. Ling, H. Liu and X. Xie, Checkable codes from group rings, arXiv: 1012.5498 (2010). Google Scholar

[19]

I. McLoughlin and T. Hurley, A group ring construction of the extended binary Golay code, IEEE Trans. Inform. Theory, 54 (2008), 4381-4383.  doi: 10.1109/TIT.2008.928260.  Google Scholar

[20]

D. S. Passman, Observations on group rings, Comm. Algebra, 5 (1977), 1119-1162.  doi: 10.1080/00927877708822213.  Google Scholar

[21]

E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN-57-103, (1957). Google Scholar

[22]

A. vom Felde, A new presentation of Cheng-Sloane's [32, 17, 8]-code, Arch. Math., 60 (1993), 508-511.  doi: 10.1007/BF01236073.  Google Scholar

show all references

References:
[1]

S. Ball, A Course in Algebraic Error-Correcting Codes, Compact Textbooks in Mathematics, 2020. doi: 10.1007/978-3-030-41153-4.  Google Scholar

[2]

M. BarbierC. Chabot and G. Quintin, On quasi-cyclic codes as a generalization of cyclic codes, Finite Fields Appl., 18 (2012), 904-919.  doi: 10.1016/j.ffa.2012.06.003.  Google Scholar

[3]

L. M. J. Bazzi and S. K. Mitter, Some randomized code constructions from group actions, IEEE Trans. Inform. Theory, 52 (2006), 3210-3219.  doi: 10.1109/TIT.2006.876244.  Google Scholar

[4]

S. D. Berman, On the theory of group codes, Cybernetics, 3 (1969), 25-31.  doi: 10.1007/BF01072842.  Google Scholar

[5]

J. J. BernalA. del Rio and J. J. Simon, An intrinsical description of group codes, Designs, Codes and Cryptography, 51 (2009), 289-300.  doi: 10.1007/s10623-008-9261-z.  Google Scholar

[6]

F. BernhardtP. Landrock and O. Manz, The extended Golay codes considered as ideals, J. Comb. Theory, Series A, 55 (1990), 235-246.  doi: 10.1016/0097-3165(90)90069-9.  Google Scholar

[7]

M. Borello, J. de la Cruz and W. Willems, On checkable codes in group algebras, Journal of Algebra and its Applications, (2021). doi: 10.1142/S0219498822501250.  Google Scholar

[8]

M. Borello and A. Jamous, Dihedral codes with prescribed minimum distance, In: Bajard J. C., Topuzoglu A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science, col 12545. Springer, Cham. (2021). doi: 10.1007/978-3-030-68869-1_8.  Google Scholar

[9]

M. BorelloP. Moree and P. Solé, Asymptotic performance of metacyclic codes, Discrete Math., 343 (2020), 111885.  doi: 10.1016/j.disc.2020.111885.  Google Scholar

[10]

M. Borello and W. Willems, Group codes over fields are asymptotically good, Finite Fields Appl., 68 (2020), 101738.  doi: 10.1016/j.ffa.2020.101738.  Google Scholar

[11]

R. C. Bose and D. K. Ray-Chaudhuri, On a class of error correcting binary group codes, Information and control, 3 (1960), 68-79.  doi: 10.1016/S0019-9958(60)90287-4.  Google Scholar

[12]

P. Charpin, Une généralisation de la construction de Berman des codes de Reed-Muller p-aire, Comm. Algebra, 16 (1988), 2231-2246.  doi: 10.1080/00927878808823689.  Google Scholar

[13]

C. L. ChenW. W. Peterson and E. J. Weldon Jr., Some results on quasi-cyclic codes, Information and control, 15 (1969), 407-423.  doi: 10.1016/S0019-9958(69)90497-5.  Google Scholar

[14]

J. H. ConwayS. J. Lomonaco Jr. and N. J. A. Sloane, A [45, 13] code with minimal distance 16, Discrete Math., 83 (1990), 213-217.  doi: 10.1016/0012-365X(90)90007-5.  Google Scholar

[15]

M. Grassl, Codetables, http://www.codetables.de/. Google Scholar

[16]

A. Hocquenghem, Codes correcteurs d'erreurs, Chiffres, 2 (1959), 147-156.   Google Scholar

[17] W. C. HuffmanJ.-L. Kim and P. Solé, Concise Encyclopedia of Coding Theory, CRC Press, 2021.  doi: 10.1201/9781315147901.  Google Scholar
[18]

S. Jitman, S. Ling, H. Liu and X. Xie, Checkable codes from group rings, arXiv: 1012.5498 (2010). Google Scholar

[19]

I. McLoughlin and T. Hurley, A group ring construction of the extended binary Golay code, IEEE Trans. Inform. Theory, 54 (2008), 4381-4383.  doi: 10.1109/TIT.2008.928260.  Google Scholar

[20]

D. S. Passman, Observations on group rings, Comm. Algebra, 5 (1977), 1119-1162.  doi: 10.1080/00927877708822213.  Google Scholar

[21]

E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN-57-103, (1957). Google Scholar

[22]

A. vom Felde, A new presentation of Cheng-Sloane's [32, 17, 8]-code, Arch. Math., 60 (1993), 508-511.  doi: 10.1007/BF01236073.  Google Scholar

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