Canonization of linear codes over \mathbb Z<SUB><I>4</I></SUB>

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May 2011, 5(2): 245-266. doi: 10.3934/amc.2011.5.245

Canonization of linear codes over $\mathbb Z$₄

Thomas Feulner ^1,

1.	Department of Mathematics, University of Bayreuth, 95440 Bayreuth

Received April 2010 Revised October 2010 Published May 2011

Abstract
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Two linear codes $C, C' \leq \mathbb Z$₄ⁿ are equivalent if there is a permutation $\pi \in S_n$ of the coordinates and a vector $\varphi \in \{1,3\}^n$ of column multiplications such that $(\varphi; \pi) C = C'$. This generalizes the notion of code equivalence of linear codes over finite fields.
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.

Keywords: canonization, group action, representative, coding theory, isometry, Automorphism group, $\mathbb Z$₄-linear code..

Mathematics Subject Classification: Primary: 05E20; Secondary: 20B25, 94B0.

Citation: Thomas Feulner. Canonization of linear codes over $\mathbb Z$₄. Advances in Mathematics of Communications, 2011, 5 (2) : 245-266. doi: 10.3934/amc.2011.5.245

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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