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, (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.  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.  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.   Google Scholar

[5]

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

[7]

R. Laue, Constructing objects up to isomorphism, simple 9-designs with small parameters,, in, (2001), 232.   Google Scholar

[8]

J. S. Leon, Computing automorphism groups of error-correcting codes,, IEEE Trans. Inform. Theory, 28 (1982), 496.  doi: 10.1109/TIT.1982.1056498.  Google Scholar

[9]

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

[10]

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

[11]

E. Petrank and R. M. Roth, Is code equivalence easy to decide?,, IEEE Trans. Inform. Theory, 43 (1997), 1602.  doi: 10.1109/18.623157.  Google Scholar

[12]

C. C. Sims, Computation with permutation groups,, in, (1971), 23.   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, (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.  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.  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.   Google Scholar

[5]

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

[7]

R. Laue, Constructing objects up to isomorphism, simple 9-designs with small parameters,, in, (2001), 232.   Google Scholar

[8]

J. S. Leon, Computing automorphism groups of error-correcting codes,, IEEE Trans. Inform. Theory, 28 (1982), 496.  doi: 10.1109/TIT.1982.1056498.  Google Scholar

[9]

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

[10]

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

[11]

E. Petrank and R. M. Roth, Is code equivalence easy to decide?,, IEEE Trans. Inform. Theory, 43 (1997), 1602.  doi: 10.1109/18.623157.  Google Scholar

[12]

C. C. Sims, Computation with permutation groups,, in, (1971), 23.   Google Scholar

[1]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[2]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
[3]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341
[4]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[5]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[6]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[7]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[8]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[9]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[10]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[11]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[12]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168
[13]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
[14]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[15]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[16]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[17]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences