Subspace Codes in PG(2N − 1, Q)

An (r,M,2δ;k)q constant-dimension subspace code, δ > 1, is a collection C of (k − 1)-dimensional projective subspaces of PG(r − 1,q) such that every (k − δ)-dimensional projective subspace of PG(r − 1,q) is contained in at most one member of C. Constant-dimension subspace codes gained recently lot of interest due to the work by Koetter and Kschischang [20], where they presented an application of such codes for error-correction in random network coding. Here a (2n,M,4;n)q constant-dimension subspace code is constructed, for every n ≥ 4. The size of our codes is considerably larger than all known constructions so far, whenever n > 4. When n = 4 a further improvement is provided by constructing an (8,M,4;4)q constant-dimension subspace code, with M = q12 + q2(q2 + 1)2(q2 + q + 1) + 1.

[1]  Albrecht Beutelspacher,et al.  Partial spreads in finite projective spaces and partial designs , 1975 .

[2]  Giuseppe Marino,et al.  Non-linear maximum rank distance codes , 2016, Des. Codes Cryptogr..

[3]  Frank R. Kschischang,et al.  Subspace Codes , 2009, IMACC.

[4]  R. Trombetti,et al.  Maximum scattered linear sets of pseudoregulus type and the Segre variety $\mathcal{S}_{n,n}$ , 2012, 1211.3604.

[5]  Alexander Vardy,et al.  Error-Correcting Codes in Projective Space , 2011, IEEE Trans. Inf. Theory.

[6]  R. H. Dye,et al.  Spreads and classes of maximal subgroups ofGLn(q),SLn(q),PGLn(q) andPSLn(q) , 1991 .

[7]  M. Hasse Dresden,et al.  H. Boerner, Darstellungen von Gruppen. 2. überarb. Auflage. (Die Grundlehren der mathematischen Wissenschaften, Band 74). XIV + 317 S. m. 14 Abb. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 58,– , 1969 .

[8]  Shu-Tao Xia,et al.  Johnson type bounds on constant dimension codes , 2007, Des. Codes Cryptogr..

[9]  J. Thas,et al.  General Galois geometries , 1992 .

[10]  Philippe Delsarte,et al.  Bilinear Forms over a Finite Field, with Applications to Coding Theory , 1978, J. Comb. Theory A.

[11]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[12]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2008, IEEE Trans. Inf. Theory.

[13]  Ron M. Roth,et al.  Author's Reply to Comments on 'Maximum-rank array codes and their application to crisscross error correction' , 1991, IEEE Trans. Inf. Theory.

[14]  B. Segre.  Forme e geometrie hermitiane, con particolare riguardo al caso finito , 1965 .

[15]  N. Gill.  Polar spaces and embeddings of classical groups , 2006, math/0603364.

[16]  Alberto Ravagnani,et al.  Subspace codes from Ferrers diagrams , 2014, ArXiv.

[17]  L. Storme,et al.  Small maximal partial spreads in classical finite polar spaces , 2010 .

[18]  Partitioning the complement of a simplex in PG(e,qd+1) into copies of PG(d,q) , 1988 .

[19]  Joachim Rosenthal,et al.  New Improvements on the Echelon-Ferrers Construction , 2010, ArXiv.

[20]  Natalia Silberstein,et al.  Error-Correcting Codes in Projective Spaces Via Rank-Metric Codes and Ferrers Diagrams , 2008, IEEE Transactions on Information Theory.

[21]  Beniamino Segre,et al.  Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane , 1964 .

[22]  Michel Lavrauw,et al.  Field reduction and linear sets in finite geometry , 2013, 1310.8522.

[23]  Sascha Kurz,et al.  Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4 , 2014 .

[24]  Antonio Cossidente,et al.  On subspace codes , 2016, Des. Codes Cryptogr..

[25]  Natalia Silberstein,et al.  Codes and designs related to lifted MRD codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[26]  B. Huppert.  Endliche Gruppen I , 1967 .

[27]  J. Hirschfeld.  Finite projective spaces of three dimensions , 1986 .

[28]  D. E. Taylor.  The geometry of the classical groups , 1992 .

[29]  Greta Panova,et al.  Matrices with restricted entries and q-analogues of permutations , 2010, 1011.4539.

[30]  Peter Wild,et al.  FINITE PROJECTIVE SPACES OF THREE DIMENSIONS (Oxford Mathematical Monographs) , 1987 .