American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Comb., 7 (1998), 116-126.  Google Scholar

[5]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.  Google Scholar

[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. Google Scholar

[7]

R. Laue, Constructing objects up to isomorphism, simple 9-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, (2001), 232-260.  Google Scholar

[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.  Google Scholar

[9]

B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324. doi: 10.1006/jagm.1997.0898.  Google Scholar

[10]

A. A. Nechaev, Kerdock's code in cyclic form, Diskret. Mat., 1 (1989), 123-139.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Comb., 7 (1998), 116-126.  Google Scholar

[5]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.  Google Scholar

[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. Google Scholar

[7]

R. Laue, Constructing objects up to isomorphism, simple 9-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, (2001), 232-260.  Google Scholar

[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.  Google Scholar

[9]

B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324. doi: 10.1006/jagm.1997.0898.  Google Scholar

[10]

A. A. Nechaev, Kerdock's code in cyclic form, Diskret. Mat., 1 (1989), 123-139.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[1]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
[2]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[3]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015
[4]

Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
[5]

Thomas Feulner. The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Advances in Mathematics of Communications, 2009, 3 (4) : 363-383. doi: 10.3934/amc.2009.3.363
[6]

Makoto Araya, Masaaki Harada, Hiroki Ito, Ken Saito. On the classification of $\mathbb{Z}_4$-codes. Advances in Mathematics of Communications, 2017, 11 (4) : 747-756. doi: 10.3934/amc.2017054
[7]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637
[8]

Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483
[9]

Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425
[10]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[11]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038
[12]

Habibul Islam, Om Prakash, Patrick Solé. $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020094
[13]

Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043
[14]

Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135
[15]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058
[16]

Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
[17]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453
[18]

Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287
[19]

Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017
[20]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences