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May  2013, 7(2): 147-160. doi: 10.3934/amc.2013.7.147

Isometry and automorphisms of constant dimension codes

Anna-Lena Trautmann 1,

1. 

Institute of Mathematics, University of Zurich, Switzerland

Received  June 2012 Published  May 2013

We define linear and semilinear isometry for general subspace codes, used for random network coding. Furthermore, some results on isometry classes and automorphism groups of known constant dimension code constructions are derived.
Keywords: Random network coding, finite fields, Grassmannian, projective space, automorphism groups., isometries, constant dimension codes.
Mathematics Subject Classification: 11T71, 14G5.
Citation: Anna-Lena Trautmann. Isometry and automorphisms of constant dimension codes. Advances in Mathematics of Communications, 2013, 7 (2) : 147-160. doi: 10.3934/amc.2013.7.147
References:
[1]

R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216. doi: 10.1109/18.850663.  Google Scholar

[2]

E. Artin, Geometric algebra, in "Interscience Tracts in Pure and Applied Mathematics,'' John Wiley & Sons, 1988.  Google Scholar

[3]

R. Baer, Linear algebra and projective geometry, in "Pure and Applied Mathematics,'' Academic Press, 1952.  Google Scholar

[4]

T. P. Berger, Isometries for rank distance and permutation group of gabidulin codes, IEEE Trans. Inform. Theory, 49 (2003), 3016-3019. doi: 10.1109/TIT.2003.819322.  Google Scholar

[5]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.  Google Scholar

[6]

T. Etzion and A. Vardy, Error-correcting codes in projective space, in "IEEE International Symposium on Information Theory,'' (2008), 871-875. Google Scholar

[7]

T. Feulner, Canonical forms and automorphisms in the projective space, preprint, 2012. Google Scholar

[8]

E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.  Google Scholar

[9]

E. Gorla, F. Manganiello and J. Rosenthal, An algebraic approach for decoding spread codes, Adv. Math. Commun., 6 (2012), 443-466. doi: 10.3934/amc.2012.6.443.  Google Scholar

[10]

J. W. P. Hirschfeld, "Finite Projective Spaces of Three Dimensions,'' The Clarendon Press, Oxford University Press, New York, 1985.  Google Scholar

[11]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[12]

J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991.  Google Scholar

[13]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in "IMA Int. Conf.,'' (2009), 1-21.  Google Scholar

[14]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in "Mathematical Methods in Computer Science,'' Springer, Berlin, (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.  Google Scholar

[15]

R. Kötter and F. R. 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

[16]

F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding, in "Proceedings of the 2008 IEEE International Symposium on Information Theory,'' Toronto, (2008), 851-855. doi: 10.1109/ISIT.2008.4595113.  Google Scholar

[17]

F. Manganiello and A.-L. Trautmann, Spread decoding in extension fields,, preprint, ().   Google Scholar

[18]

D. Silva and F. R. Kschischang, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, 55 (2009), 5479-5490. doi: 10.1109/TIT.2009.2032817.  Google Scholar

[19]

D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.  Google Scholar

[20]

V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.  Google Scholar

[21]

A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes,, preprint, ().   Google Scholar

[22]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in "IEEE Information Theory Workshop (ITW),'' Dublin, (2010), 1-4. Google Scholar

show all references

References:
[1]

R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216. doi: 10.1109/18.850663.  Google Scholar

[2]

E. Artin, Geometric algebra, in "Interscience Tracts in Pure and Applied Mathematics,'' John Wiley & Sons, 1988.  Google Scholar

[3]

R. Baer, Linear algebra and projective geometry, in "Pure and Applied Mathematics,'' Academic Press, 1952.  Google Scholar

[4]

T. P. Berger, Isometries for rank distance and permutation group of gabidulin codes, IEEE Trans. Inform. Theory, 49 (2003), 3016-3019. doi: 10.1109/TIT.2003.819322.  Google Scholar

[5]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.  Google Scholar

[6]

T. Etzion and A. Vardy, Error-correcting codes in projective space, in "IEEE International Symposium on Information Theory,'' (2008), 871-875. Google Scholar

[7]

T. Feulner, Canonical forms and automorphisms in the projective space, preprint, 2012. Google Scholar

[8]

E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.  Google Scholar

[9]

E. Gorla, F. Manganiello and J. Rosenthal, An algebraic approach for decoding spread codes, Adv. Math. Commun., 6 (2012), 443-466. doi: 10.3934/amc.2012.6.443.  Google Scholar

[10]

J. W. P. Hirschfeld, "Finite Projective Spaces of Three Dimensions,'' The Clarendon Press, Oxford University Press, New York, 1985.  Google Scholar

[11]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[12]

J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991.  Google Scholar

[13]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in "IMA Int. Conf.,'' (2009), 1-21.  Google Scholar

[14]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in "Mathematical Methods in Computer Science,'' Springer, Berlin, (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.  Google Scholar

[15]

R. Kötter and F. R. 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

[16]

F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding, in "Proceedings of the 2008 IEEE International Symposium on Information Theory,'' Toronto, (2008), 851-855. doi: 10.1109/ISIT.2008.4595113.  Google Scholar

[17]

F. Manganiello and A.-L. Trautmann, Spread decoding in extension fields,, preprint, ().   Google Scholar

[18]

D. Silva and F. R. Kschischang, On metrics for error correction in network coding, IEEE Trans. Inform. Theory, 55 (2009), 5479-5490. doi: 10.1109/TIT.2009.2032817.  Google Scholar

[19]

D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967. doi: 10.1109/TIT.2008.928291.  Google Scholar

[20]

V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.  Google Scholar

[21]

A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes,, preprint, ().   Google Scholar

[22]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in "IEEE Information Theory Workshop (ITW),'' Dublin, (2010), 1-4. Google Scholar

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