On lattices, binary codes, and network codes

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May 2011, 5(2): 225-232. doi: 10.3934/amc.2011.5.225

On lattices, binary codes, and network codes

Michael Braun ^1,

1.	University of Applied Sciences Darmstadt, Faculty of Computer Science, D-64295 Darmstadt, Germany

Received April 2010 Revised October 2010 Published May 2011

Abstract
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Network codes are sets of subspaces of a finite vectorspace over a finite field. Recently, this class of codes has found application in the error correction of message transmission within networks. Furthermore, binary codes can be represented as sets of subsets of a finite set. Hence, both kinds of codes can be regarded as substructures of lattices — in the first case it is the linear lattice and in the second case it is the power set lattice. This observation leads us to a more general investigation of similarities of both theories by means of lattice theory. In this paper we first examine general results of lattices in order to comprise basic considerations of network coding and binary vector coding theory. Afterwards we consider the issue of finding complements of subspaces.

Keywords: posets, complementary spaces., lattices, metric spaces, Binary codes, network codes.

Mathematics Subject Classification: 03G1.

Citation: Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225

References:

[1]	G. Birkhoff, "Lattice Theory,'' revised edition, Amer. Math. Soc. Colloquium Publications, 1948.
[2]	W. E. Clark, Matching subspaces with complements in finite vector spaces, in "Bulletin of the Inst. of Combinatorics and its Applications,'' 8 (1992), 33-38.
[3]	T. Etzion and A. Vardy, Coding theory in projective spaces, in "Information Theory and Applications Workshop,'' San Diego, USA, (2008).
[4]	T. Etzion and A. Vardy, Error-correcting codes in projective spaces, in "IEEE International Symposium on Information Theory,'' Toronto, Canada, (2008). doi: 10.1109/ISIT.2008.4595111.
[5]	R. Koetter and F. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

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References:

[1]	G. Birkhoff, "Lattice Theory,'' revised edition, Amer. Math. Soc. Colloquium Publications, 1948.
[2]	W. E. Clark, Matching subspaces with complements in finite vector spaces, in "Bulletin of the Inst. of Combinatorics and its Applications,'' 8 (1992), 33-38.
[3]	T. Etzion and A. Vardy, Coding theory in projective spaces, in "Information Theory and Applications Workshop,'' San Diego, USA, (2008).
[4]	T. Etzion and A. Vardy, Error-correcting codes in projective spaces, in "IEEE International Symposium on Information Theory,'' Toronto, Canada, (2008). doi: 10.1109/ISIT.2008.4595111.
[5]	R. Koetter and F. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

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