Optimal binary subspace codes of length 6, constant dimension 3 and minimum 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 $\operatorname{PG}(5,2)$ mutually intersecting in at most a point is $77$. Optimal binary $(v,M,d;k)=(6,77,4;3)$ subspace codes are classified into $5$ isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any $q$, yielding a new family of $q$-ary $(6,q^6+2q^2+2q+1,4;3)$ subspace codes.

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

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

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

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

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

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

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

[8]  Frank R. Kschischang,et al.  A Rank-Metric Approach to Error Control in Random Network Coding , 2007, IEEE Transactions on Information Theory.

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

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

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

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

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

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

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

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

[17]  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.

[18]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2007, IEEE Transactions on Information Theory.

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

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