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Algebraic structure of the minimal support codewords set of some linear codes
Canonization of linear codes over $\mathbb Z$4
1. | Department of Mathematics, University of Bayreuth, 95440 Bayreuth |
In a previous paper, the author has described an algorithm to compute the canonical form of a linear code over a finite field. In the present paper, an algorithm is presented to compute the canonical form as well as the automorphism group of a linear code over $\mathbb Z$4. This solves the isomorphism problem for $\mathbb Z$4-linear codes. An efficient implementation of this algorithm is described and some results on the classification of linear codes over $\mathbb Z$4 for small parameters are discussed.
References:
[1] |
A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, "Error-Correcting Linear Codes, Classification by Isometry and Applications,'' Springer, Berlin, 2006. |
[2] |
T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Adv. Math. Commun., 3 (2009), 363-383.
doi: 10.3934/amc.2009.3.363. |
[3] |
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preperata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[4] |
T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Comb., 7 (1998), 116-126. |
[5] |
W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. |
[6] |
M. Kiermaier and J. Zwanzger, A $\mathbbZ_4$-linear code of high minimum Lee distance derived from a hyperoval, Adv. Math. Commun., 5 (2011), 275-286. |
[7] |
R. Laue, Constructing objects up to isomorphism, simple 9-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, (2001), 232-260. |
[8] |
J. S. Leon, Computing automorphism groups of error-correcting codes, IEEE Trans. Inform. Theory, 28 (1982), 496-511.
doi: 10.1109/TIT.1982.1056498. |
[9] |
B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324.
doi: 10.1006/jagm.1997.0898. |
[10] |
A. A. Nechaev, Kerdock's code in cyclic form, Diskret. Mat., 1 (1989), 123-139. |
[11] |
E. Petrank and R. M. Roth, Is code equivalence easy to decide?, IEEE Trans. Inform. Theory, 43 (1997), 1602-1604.
doi: 10.1109/18.623157. |
[12] |
C. C. Sims, Computation with permutation groups, in "Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation, SYMSAC '71,'' (1971), 23-28. |
show all references
References:
[1] |
A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, "Error-Correcting Linear Codes, Classification by Isometry and Applications,'' Springer, Berlin, 2006. |
[2] |
T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Adv. Math. Commun., 3 (2009), 363-383.
doi: 10.3934/amc.2009.3.363. |
[3] |
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preperata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[4] |
T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Comb., 7 (1998), 116-126. |
[5] |
W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. |
[6] |
M. Kiermaier and J. Zwanzger, A $\mathbbZ_4$-linear code of high minimum Lee distance derived from a hyperoval, Adv. Math. Commun., 5 (2011), 275-286. |
[7] |
R. Laue, Constructing objects up to isomorphism, simple 9-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, (2001), 232-260. |
[8] |
J. S. Leon, Computing automorphism groups of error-correcting codes, IEEE Trans. Inform. Theory, 28 (1982), 496-511.
doi: 10.1109/TIT.1982.1056498. |
[9] |
B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324.
doi: 10.1006/jagm.1997.0898. |
[10] |
A. A. Nechaev, Kerdock's code in cyclic form, Diskret. Mat., 1 (1989), 123-139. |
[11] |
E. Petrank and R. M. Roth, Is code equivalence easy to decide?, IEEE Trans. Inform. Theory, 43 (1997), 1602-1604.
doi: 10.1109/18.623157. |
[12] |
C. C. Sims, Computation with permutation groups, in "Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation, SYMSAC '71,'' (1971), 23-28. |
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