American Institute of Mathematical Sciences

Advanced
November  2007, 1(4): 477-487. doi: 10.3934/amc.2007.1.477

Decoding the Mathieu group M12

Robert F. Bailey 1, and  John N. Bray 2,

1. 

School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada
2. 

School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, United Kingdom

Received  June 2007 Revised  October 2007 Published  October 2007

The sporadic Mathieu group M12 can be viewed as an error-correcting code, where the codewords are the group's elements written as permutations in list form, and with the usual Hamming distance. We investigate the properties of this group as a code, in particular determining completely the probabilities of successful and ambiguous decoding of words with more than 3 errors (which is the number that can be guaranteed to be corrected).
Keywords: Permutation code, Mathieu group M12..
Mathematics Subject Classification: Primary: 94B60; Secondary: 20D08, 94B2.
Citation: Robert F. Bailey, John N. Bray. Decoding the Mathieu group M12. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477
[1]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[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]

Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
[4]

Shiping Cao, Anthony Coniglio, Xueyan Niu, Richard H. Rand, Robert S. Strichartz. The mathieu differential equation and generalizations to infinite fractafolds. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1795-1845. doi: 10.3934/cpaa.2020073
[5]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[6]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[7]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[8]

Nikolay Yankov. Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7. Advances in Mathematics of Communications, 2014, 8 (1) : 73-81. doi: 10.3934/amc.2014.8.73
[9]

Armando G. M. Neves. Upper and lower bounds on Mathieu characteristic numbers of integer orders. Communications on Pure & Applied Analysis, 2004, 3 (3) : 447-464. doi: 10.3934/cpaa.2004.3.447
[10]

Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431
[11]

Ethel Mokotoff. Algorithms for bicriteria minimization in the permutation flow shop scheduling problem. Journal of Industrial & Management Optimization, 2011, 7 (1) : 253-282. doi: 10.3934/jimo.2011.7.253
[12]

Ricardo P. Beausoleil, Rodolfo A. Montejo. A study with neighborhood searches to deal with multiobjective unconstrained permutation problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 193-216. doi: 10.3934/jimo.2009.5.193
[13]

Amin Sakzad, Mohammad-Reza Sadeghi, Daniel Panario. Cycle structure of permutation functions over finite fields and their applications. Advances in Mathematics of Communications, 2012, 6 (3) : 347-361. doi: 10.3934/amc.2012.6.347
[14]

Nian Li, Qiaoyu Hu. A conjecture on permutation trinomials over finite fields of characteristic two. Advances in Mathematics of Communications, 2019, 13 (3) : 505-512. doi: 10.3934/amc.2019031
[15]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[16]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[17]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010
[18]

O. A. Veliev. On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1537-1562. doi: 10.3934/cpaa.2020077
[19]

Julio C. Rebelo, Ana L. Silva. On the Burnside problem in Diff(M). Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 423-439. doi: 10.3934/dcds.2007.17.423
[20]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences