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

Isometry and automorphisms of constant dimension codes

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.
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. doi: 10.1109/18.850663.

[2]

E. Artin, Geometric algebra,, in, (1988).

[3]

R. Baer, Linear algebra and projective geometry,, in, (1952).

[4]

T. P. Berger, Isometries for rank distance and permutation group of gabidulin codes,, IEEE Trans. Inform. Theory, 49 (2003), 3016. 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. doi: 10.1109/TIT.2009.2021376.

[6]

T. Etzion and A. Vardy, Error-correcting codes in projective space,, in, (2008), 871.

[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.

[9]

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

[10]

J. W. P. Hirschfeld, "Finite Projective Spaces of Three Dimensions,'', The Clarendon Press, (1985).

[11]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition,, The Clarendon Press, (1998).

[12]

J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'', The Clarendon Press, (1991).

[13]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes,, in, (2009), 1.

[14]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance,, in, (2008), 31. 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. doi: 10.1109/TIT.2008.926449.

[16]

F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding,, in, (2008), 851. doi: 10.1109/ISIT.2008.4595113.

[17]

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

[18]

D. Silva and F. R. Kschischang, On metrics for error correction in network coding,, IEEE Trans. Inform. Theory, 55 (2009), 5479. 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. doi: 10.1109/TIT.2008.928291.

[20]

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

[21]

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

[22]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding,, in, (2010), 1.

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. doi: 10.1109/18.850663.

[2]

E. Artin, Geometric algebra,, in, (1988).

[3]

R. Baer, Linear algebra and projective geometry,, in, (1952).

[4]

T. P. Berger, Isometries for rank distance and permutation group of gabidulin codes,, IEEE Trans. Inform. Theory, 49 (2003), 3016. 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. doi: 10.1109/TIT.2009.2021376.

[6]

T. Etzion and A. Vardy, Error-correcting codes in projective space,, in, (2008), 871.

[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.

[9]

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

[10]

J. W. P. Hirschfeld, "Finite Projective Spaces of Three Dimensions,'', The Clarendon Press, (1985).

[11]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition,, The Clarendon Press, (1998).

[12]

J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'', The Clarendon Press, (1991).

[13]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes,, in, (2009), 1.

[14]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance,, in, (2008), 31. 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. doi: 10.1109/TIT.2008.926449.

[16]

F. Manganiello, E. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding,, in, (2008), 851. doi: 10.1109/ISIT.2008.4595113.

[17]

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

[18]

D. Silva and F. R. Kschischang, On metrics for error correction in network coding,, IEEE Trans. Inform. Theory, 55 (2009), 5479. 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. doi: 10.1109/TIT.2008.928291.

[20]

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

[21]

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

[22]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding,, in, (2010), 1.

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