Optimal Binary Subspace Codes of Length 6 , Constant Dimension 3 and Minimum Subspace Distance 4

It is shown that the maximum size of a binary subspace code of packet length v = 6, minimum subspace distance d = 4, and constant dimension k = 3 is M = 77; in Finite Geometry terms, the maximum number of planes in PG(5; 2) mutually intersecting in at most a point is 77. Optimal binary (v;M; d; k) = (6; 77; 4; 3) subspace codes are classi ed into 5 isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and nite elds theory and generalizes to any q, yielding a new family of q-ary (6; q + 2q + 2q + 1; 4; 3) subspace codes.

[1]  Nicola Melone,et al.  A characterization of Grassmann and attenuated spaces as (0, alpha)-geometries , 2003, Eur. J. Comb..

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

[3]  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.

[4]  Classi cation of partial spreads in PG ( 4 ; 2 ) , 2004 .

[5]  Z. Wan,et al.  Geometry of Matrices , 1996 .

[6]  Thomas Honold,et al.  Good random matrices over finite fields , 2010, Adv. Math. Commun..

[7]  D. Knuth.  Subspaces, subsets, and partitions , 1971 .

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

[9]  Tuvi Etzion,et al.  Problems on q-Analogs in Coding Theory , 2013, ArXiv.

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

[11]  Frank R. Kschischang,et al.  An Introduction to Network Coding , 2012 .

[12]  Sascha Kurz,et al.  Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance , 2008, MMICS.

[13]  H. Maldeghem,et al.  Some Classes of Rank 2 Geometries , 1995 .

[14]  Thomas Feulner.  Canonical Forms and Automorphisms in the Projective Space , 2013, ArXiv.

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

[16]  Thomas Feulner.  The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes , 2009, Adv. Math. Commun..

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

[18]  Ronald Shaw.  Subsets of PG(n,2) and Maximal Partial Spreads in PG(4,2) , 2000, Des. Codes Cryptogr..

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