Abstract
In this paper we construct constant dimension codes with prescribed minimum distance. There is an increased interest in subspace codes in general since a paper [13] by Kötter and Kschischang where they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to construct constant dimension codes. Using this approach we found many new constant dimension codes with a larger number of codewords than previously known codes. We finally give a table of the best constant dimension codes we found.
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References
Ahlswede, R., Aydinian, H.K., Khachatrian, L.H.: On perfect codes and related concepts. Des. Codes Cryptography 22(3), 221–237 (2001)
Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-correcting linear codes. Classification by isometry and applications. With CD-ROM. In: Algorithms and Computation in Mathematics 18, p. xxix, 798. Springer, Berlin (2006)
Betten, A., Kerber, A., Kohnert, A., Laue, R., Wassermann, A.: The discovery of simple 7-designs with automorphism group PΓ L(2,32). In: Giusti, M., Cohen, G., Mora, T. (eds.) AAECC 1995. LNCS, vol. 948, pp. 131–145. Springer, Heidelberg (1995)
Braun, M.: Construction of linear codes with large minimum distance. IEEE Transactions on Information Theory 50(8), 1687–1691 (2004)
Braun, M., Kohnert, A., Wassermann, A.: Optimal linear codes from matrix groups. IEEE Transactions on Information Theory 51(12), 4247–4251 (2005)
Braun, M.: Some new designs over finite fields. Bayreuther Math. Schr. 74, 58–68 (2005)
Braun, M., Kerber, A., Laue, R.: Systematic construction of q-analogs of t-(v,k,λ)-designs. Des. Codes Cryptography 34(1), 55–70 (2005)
Braun, M., Kohnert, A., Wassermann, A.: Construction of (n,r)-arcs in PG(2,q). Innov. Incidence Geom. 1, 133–141 (2005)
Drudge, K.: On the orbits of Singer groups and their subgroups. Electronic Journal Comb. 9(1), 10 p. (2002)
Etzion, T., Silberstein, N.: Construction of error-correcting codes for random network coding (submitted, 2008) (in arXiv 0805.3528)
Etzion, T., Vardy, A.: Error-Correcting codes in projective space. In: ISIT Proceedings, 5 p. (2008)
Gadouleau, M., Yan, Z.: Constant-rank codes and their connection to constant-dimension codes (submitted, 2008) (in arXiv 0803.2262)
Kötter, R., Kschischang, F.: Coding for errors and erasures in random network coding. IEEE Transactions on Information Theory 54(8), 3579–3391 (2008)
Kramer, E.S., Mesner, D.M.: t-designs on hypergraphs. Discrete Math. 15, 263–296 (1976)
Maruta, T., Shinohara, M., Takenaka, M.: Constructing linear codes from some orbits of projectivities. Discrete Math. 308(5-6), 832–841 (2008)
Niskanen, S., Östergård, P.R.J.: Cliquer user’s guide, version 1.0. Technical Report T48, Communications Laboratory, Helsinki University of Technology, Espoo, Finland (2003)
Schwartz, M., Etzion, T.: Codes and anticodes in the Grassman graph. J. Comb. Theory, Ser. A 97(1), 27–42 (2002)
Silberstein, N.: Coding theory in projective space. Ph.D. proposal (2008) (in arXiv 0805.3528)
Thomas, S.: Designs over finite fields. Geom. Dedicata 24, 237–242 (1987)
Thomas, S.: Designs and partial geometries over finite fields. Geom. Dedicata 63(3), 247–253 (1996)
Tonchev, V.D.: Quantum codes from caps. Discrete Math. (to appear, 2008)
Wassermann, A.: Lattice point enumeration and applications. Bayreuther Math. Schr. 73, 1–114 (2006)
Xia, S.-T., Fu, F.-W.: Johnson type bounds on constant dimension codes (submitted, 2007) (in arXiv 0709.1074)
Zwanzger, J.: A heuristic algorithm for the construction of good linear codes. IEEE Transactions on Information Theory 54(5), 2388–2392 (2008)
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Kohnert, A., Kurz, S. (2008). Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance. In: Calmet, J., Geiselmann, W., Müller-Quade, J. (eds) Mathematical Methods in Computer Science. Lecture Notes in Computer Science, vol 5393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89994-5_4
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DOI: https://doi.org/10.1007/978-3-540-89994-5_4
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