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

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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

Abstract
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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), 393-402. doi: 10.3934/amc.2008.2.393.
[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), 2984-3002. doi: 10.1016/j.disc.2007.08.024.
[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), 1194-1205. doi: 10.1109/18.761269.
[4]	A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35. doi: 10.1007/BF01390768.
[5]	W. Ebeling, "Lattices and Codes,'' Friedr. Vieweg & Sohn, Braunschweig, 1994.
[6]	Y. H. Park, Modular independence and generator matrices for codes over $\mathbb Z_m$, Des. Codes Crypt., 50 (2009), 147-162. doi: 10.1007/s10623-008-9220-8.
[7]	H. Petersson, "Modulfunktionen und Quadratische Formen,'' Springer-Verlag, Berlin, 1982.

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References:

[1]	B. van Asch and F. Martens, Lee weight enumerators of self-dual codes and theta functions, Adv. Math. Commun., 2 (2008), 393-402. doi: 10.3934/amc.2008.2.393.
[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), 2984-3002. doi: 10.1016/j.disc.2007.08.024.
[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), 1194-1205. doi: 10.1109/18.761269.
[4]	A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35. doi: 10.1007/BF01390768.
[5]	W. Ebeling, "Lattices and Codes,'' Friedr. Vieweg & Sohn, Braunschweig, 1994.
[6]	Y. H. Park, Modular independence and generator matrices for codes over $\mathbb Z_m$, Des. Codes Crypt., 50 (2009), 147-162. doi: 10.1007/s10623-008-9220-8.
[7]	H. Petersson, "Modulfunktionen und Quadratische Formen,'' Springer-Verlag, Berlin, 1982.

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