American Institute of Mathematical Sciences

Advanced
February  2012, 6(1): 65-68. doi: 10.3934/amc.2012.6.65

A note on the minimum Lee distance of certain self-dual modular codes

Bram van Asch 1, and  Frans Martens 1,

1. 

Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven

Received  September 2010 Revised  October 2011 Published  January 2012

In a former paper we investigated the connection between $p$-ary linear codes, $p$ prime, and theta functions. Corresponding to a given code a suitable lattice and its associated theta function were defined. Using results from the theory of modular forms we got an algorithm to determine an upper bound for the minimum Lee distance of certain self-dual codes. In this note we generalize this result to $m$-ary codes, where $m$ is either a power of a prime, or $m$ is square-free. If $m$ is of a different form the generalization will not work. A class of examples to illustrate this fact is given.
Keywords: modular form, Lee weight enumerator, self-dual code, theta function, lattice, Modular code, minimum Lee distance..
Mathematics Subject Classification: Primary: 11F27, 11H71, 94B05; Secondary: 11F11, 94B6.
Citation: Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65
References:
[1]

B. van Asch and F. Martens, Lee weight enumerators of self-dual codes and theta functions,, Adv. Math. Commun., 2 (2008), 393.  doi: 10.3934/amc.2008.2.393.  Google Scholar

[2]

J. M. P. Balmaceda, R. A. L. Betty and F. R. Nemenzo, Mass formula for self-dual codes over $\mathbb Z$p2,, Discr. Math., 308 (2008), 2984.  doi: 10.1016/j.disc.2007.08.024.  Google Scholar

[3]

E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices, and invariant rings,, IEEE Trans. Inform. Theory, 45 (1999), 1194.  doi: 10.1109/18.761269.  Google Scholar

[4]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes,, Des. Codes Crypt., 6 (1995), 21.  doi: 10.1007/BF01390768.  Google Scholar

[5]

W. Ebeling, "Lattices and Codes,'', Friedr. Vieweg & Sohn, (1994).   Google Scholar

[6]

Y. H. Park, Modular independence and generator matrices for codes over $\mathbb Z_m$,, Des. Codes Crypt., 50 (2009), 147.  doi: 10.1007/s10623-008-9220-8.  Google Scholar

[7]

H. Petersson, "Modulfunktionen und Quadratische Formen,'', Springer-Verlag, (1982).   Google Scholar

show all references

References:
[1]

B. van Asch and F. Martens, Lee weight enumerators of self-dual codes and theta functions,, Adv. Math. Commun., 2 (2008), 393.  doi: 10.3934/amc.2008.2.393.  Google Scholar

[2]

J. M. P. Balmaceda, R. A. L. Betty and F. R. Nemenzo, Mass formula for self-dual codes over $\mathbb Z$p2,, Discr. Math., 308 (2008), 2984.  doi: 10.1016/j.disc.2007.08.024.  Google Scholar

[3]

E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices, and invariant rings,, IEEE Trans. Inform. Theory, 45 (1999), 1194.  doi: 10.1109/18.761269.  Google Scholar

[4]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes,, Des. Codes Crypt., 6 (1995), 21.  doi: 10.1007/BF01390768.  Google Scholar

[5]

W. Ebeling, "Lattices and Codes,'', Friedr. Vieweg & Sohn, (1994).   Google Scholar

[6]

Y. H. Park, Modular independence and generator matrices for codes over $\mathbb Z_m$,, Des. Codes Crypt., 50 (2009), 147.  doi: 10.1007/s10623-008-9220-8.  Google Scholar

[7]

H. Petersson, "Modulfunktionen und Quadratische Formen,'', Springer-Verlag, (1982).   Google Scholar

[1]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[2]

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
[3]

Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035
[4]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[5]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[6]

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
[7]

Masaaki Harada. New doubly even self-dual codes having minimum weight 20. Advances in Mathematics of Communications, 2020, 14 (1) : 89-96. doi: 10.3934/amc.2020007
[8]

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
[9]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[10]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[11]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[12]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[13]

Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002
[14]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[15]

Manish K. Gupta, Chinnappillai Durairajan. On the covering radius of some modular codes. Advances in Mathematics of Communications, 2014, 8 (2) : 129-137. doi: 10.3934/amc.2014.8.129
[16]

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
[17]

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
[18]

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
[19]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281
[20]

Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences