Isometry and automorphisms of constant dimension codes

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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

Abstract
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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.
[2]	E. Artin, Geometric algebra, in "Interscience Tracts in Pure and Applied Mathematics,'' John Wiley & Sons, 1988.
[3]	R. Baer, Linear algebra and projective geometry, in "Pure and Applied Mathematics,'' Academic Press, 1952.
[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.
[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.
[6]	T. Etzion and A. Vardy, Error-correcting codes in projective space, in "IEEE International Symposium on Information Theory,'' (2008), 871-875.
[7]	T. Feulner, Canonical forms and automorphisms in the projective space, preprint, 2012.
[8]	E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.
[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.
[10]	J. W. P. Hirschfeld, "Finite Projective Spaces of Three Dimensions,'' The Clarendon Press, Oxford University Press, New York, 1985.
[11]	J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2^nd edition, The Clarendon Press, Oxford University Press, New York, 1998.
[12]	J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991.
[13]	A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in "IMA Int. Conf.,'' (2009), 1-21.
[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.
[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.
[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.
[17]	F. Manganiello and A.-L. Trautmann, Spread decoding in extension fields, preprint, arXiv:1108.5881
[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.
[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.
[20]	V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.
[21]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, preprint, arXiv:1112.1238
[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.

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.
[2]	E. Artin, Geometric algebra, in "Interscience Tracts in Pure and Applied Mathematics,'' John Wiley & Sons, 1988.
[3]	R. Baer, Linear algebra and projective geometry, in "Pure and Applied Mathematics,'' Academic Press, 1952.
[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.
[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.
[6]	T. Etzion and A. Vardy, Error-correcting codes in projective space, in "IEEE International Symposium on Information Theory,'' (2008), 871-875.
[7]	T. Feulner, Canonical forms and automorphisms in the projective space, preprint, 2012.
[8]	E. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.
[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.
[10]	J. W. P. Hirschfeld, "Finite Projective Spaces of Three Dimensions,'' The Clarendon Press, Oxford University Press, New York, 1985.
[11]	J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2^nd edition, The Clarendon Press, Oxford University Press, New York, 1998.
[12]	J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991.
[13]	A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in "IMA Int. Conf.,'' (2009), 1-21.
[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.
[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.
[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.
[17]	F. Manganiello and A.-L. Trautmann, Spread decoding in extension fields, preprint, arXiv:1108.5881
[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.
[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.
[20]	V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.
[21]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, preprint, arXiv:1112.1238
[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.

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