American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[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.  Google Scholar

[3]

T. Etzion and A. Vardy, Coding theory in projective spaces, in "Information Theory and Applications Workshop,'' San Diego, USA, (2008). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. Birkhoff, "Lattice Theory,'' revised edition, Amer. Math. Soc. Colloquium Publications, 1948.  Google Scholar

[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.  Google Scholar

[3]

T. Etzion and A. Vardy, Coding theory in projective spaces, in "Information Theory and Applications Workshop,'' San Diego, USA, (2008). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Daniel Heinlein, Sascha Kurz. Binary subspace codes in small ambient spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 817-839. doi: 10.3934/amc.2018048
[2]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[3]

B. K. Dass, Namita Sharma, Rashmi Verma. Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 629-639. doi: 10.3934/amc.2018037
[4]

Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773
[5]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119
[6]

Daniele Bartoli, Matteo Bonini, Massimo Giulietti. Constant dimension codes from Riemann-Roch spaces. Advances in Mathematics of Communications, 2017, 11 (4) : 705-713. doi: 10.3934/amc.2017051
[7]

Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano. Orbit codes from forms on vector spaces over a finite field. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020105
[8]

Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329
[9]

Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure & Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103
[10]

Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733
[11]

Claude Carlet, Sylvain Guilley. Complementary dual codes for counter-measures to side-channel attacks. Advances in Mathematics of Communications, 2016, 10 (1) : 131-150. doi: 10.3934/amc.2016.10.131
[12]

Finley Freibert. The classification of complementary information set codes of lengths $14$ and $16$. Advances in Mathematics of Communications, 2013, 7 (3) : 267-278. doi: 10.3934/amc.2013.7.267
[13]

Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012
[14]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025
[15]

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
[16]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
[17]

Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017
[18]

Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049
[19]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629
[20]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences