# American Institute of Mathematical Sciences

May  2011, 5(2): 177-189. doi: 10.3934/amc.2011.5.177

## Large constant dimension codes and lexicodes

 1 Computer Science Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel

Received  March 2010 Revised  September 2010 Published  May 2011

Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of $\mathbb F$qn with a given dimension. A computer search for large constant dimension codes is usually inefficient since the search space domain is extremely large. Even so, we found that some constant dimension lexicodes are larger than other known codes. We show how to make the computer search more efficient. In this context we present a formula for the computation of the distance between two subspaces, not necessarily of the same dimension.
Citation: Natalia Silberstein, Tuvi Etzion. Large constant dimension codes and lexicodes. Advances in Mathematics of Communications, 2011, 5 (2) : 177-189. doi: 10.3934/amc.2011.5.177
##### References:
 [1] G. E. Andrews and K. Eriksson, "Integer Partitions,'' Cambridge University Press, 2004.  Google Scholar [2] J. H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Trans. Inform. Theory, 32 (1986), 337-348. doi: 10.1109/TIT.1986.1057187.  Google Scholar [3] P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.  Google Scholar [4] T. Etzion and N. Silberstein, Error-correcting codes in projective space via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.  Google Scholar [5] T. Etzion and A. Vardy, Error-correcting codes in projective space, in "Proceedings of International Symposium on Information Theory,'' (2008), 871-875. Google Scholar [6] T. Etzion and A. Vardy, On $q$-analogs for Steiner systems and covering designs, Adv. Math. Commun., 5 (2011), 161-176.  Google Scholar [7] W. Fulton, "Young Tableaux,'' Cambridge University Press, 1997.  Google Scholar [8] E. M. Gabidulin, Theory of codes with maximal rank distance, Probl. Inform. Transm., 21 (1985), 1-12.  Google Scholar [9] M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216. doi: 10.1109/TIT.2010.2048447.  Google Scholar [10] R. Koetter 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 [11] A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, Lecture Notes Comp. Sci., 5393 (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.  Google Scholar [12] V. L. Levenshtein, A class of systematic codes, Soviet Math. Dokl., 1 (1960), 368-371.  Google Scholar [13] J. H. van Lint and R. M. Wilson, "A Course in Combinatorics,'' 2nd edition, Cambridge University Press, 2001.  Google Scholar [14] R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory, 37 (1991), 328-336. doi: 10.1109/18.75248.  Google Scholar [15] N. Silberstein and T. Etzion, Enumerative coding for Grassmannian space, IEEE Trans. Inform. Theory, 57 (2011), 365-374. doi: 10.1109/TIT.2010.2090252.  Google Scholar [16] D. Silva, F. R. Kschischang and R. Koetter, 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 [17] V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.  Google Scholar [18] R. P. Stanley, "Enumerative Combinatorics,'' Wadsworth, 1986. Google Scholar [19] A. J. Van Zanten, Lexicographic order and linearity, Des. Codes Crypt., 10 (1997), 85-97. doi: 10.1023/A:1008244404559.  Google Scholar

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##### References:
 [1] G. E. Andrews and K. Eriksson, "Integer Partitions,'' Cambridge University Press, 2004.  Google Scholar [2] J. H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Trans. Inform. Theory, 32 (1986), 337-348. doi: 10.1109/TIT.1986.1057187.  Google Scholar [3] P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.  Google Scholar [4] T. Etzion and N. Silberstein, Error-correcting codes in projective space via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919. doi: 10.1109/TIT.2009.2021376.  Google Scholar [5] T. Etzion and A. Vardy, Error-correcting codes in projective space, in "Proceedings of International Symposium on Information Theory,'' (2008), 871-875. Google Scholar [6] T. Etzion and A. Vardy, On $q$-analogs for Steiner systems and covering designs, Adv. Math. Commun., 5 (2011), 161-176.  Google Scholar [7] W. Fulton, "Young Tableaux,'' Cambridge University Press, 1997.  Google Scholar [8] E. M. Gabidulin, Theory of codes with maximal rank distance, Probl. Inform. Transm., 21 (1985), 1-12.  Google Scholar [9] M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216. doi: 10.1109/TIT.2010.2048447.  Google Scholar [10] R. Koetter 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 [11] A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, Lecture Notes Comp. Sci., 5393 (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.  Google Scholar [12] V. L. Levenshtein, A class of systematic codes, Soviet Math. Dokl., 1 (1960), 368-371.  Google Scholar [13] J. H. van Lint and R. M. Wilson, "A Course in Combinatorics,'' 2nd edition, Cambridge University Press, 2001.  Google Scholar [14] R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory, 37 (1991), 328-336. doi: 10.1109/18.75248.  Google Scholar [15] N. Silberstein and T. Etzion, Enumerative coding for Grassmannian space, IEEE Trans. Inform. Theory, 57 (2011), 365-374. doi: 10.1109/TIT.2010.2090252.  Google Scholar [16] D. Silva, F. R. Kschischang and R. Koetter, 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 [17] V. Skachek, Recursive code construction for random networks, IEEE Trans. Inform. Theory, 56 (2010), 1378-1382. doi: 10.1109/TIT.2009.2039163.  Google Scholar [18] R. P. Stanley, "Enumerative Combinatorics,'' Wadsworth, 1986. Google Scholar [19] A. J. Van Zanten, Lexicographic order and linearity, Des. Codes Crypt., 10 (1997), 85-97. doi: 10.1023/A:1008244404559.  Google Scholar
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