Equidistant codes in the Grassmannian

Equidistant codes over vector spaces are considered. For k -dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Plucker embedding, for 1-intersecting codes of k -dimensional subspaces over F q n , n ? ( k + 1 2 ) , where the code size is q k + 1 - 1 q - 1 is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size n i? ( n 2 ) over F q , rank n - 1 , and rank distance n - 1 .

[1]  Peter Frankl,et al.  Every large set of equidistant (0, +1, −1)-vectors forms a sunflower , 1981, Comb..

[2]  Peter Frankl,et al.  The Erdös-Ko-Rado theorem for vector spaces , 1986, J. Comb. Theory, Ser. A.

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

[4]  Michel Deza.  Une propriete extremale des plans projectifs finis dans une classe de codes equidistants , 1973, Discret. Math..

[5]  Jonathan I. Hall,et al.  Bounds for equidistant codes and partial projective planes , 1977, Discret. Math..

[6]  Torleiv Kløve,et al.  On Equidistant Constant Weight Codes , 2003, Discret. Appl. Math..

[7]  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.

[8]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

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

[10]  Johan P. Hansen,et al.  Osculating Spaces of Varieties and Linear Network Codes , 2012, CAI.

[11]  Joachim Rosenthal,et al.  On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes , 2014, Des. Codes Cryptogr..

[12]  Alexander Vardy,et al.  Linearity and Complements in Projective Space , 2011, ArXiv.

[13]  Johan P. Hansen.  Forms and Linear Network Codes , 2013, ArXiv.

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

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

[16]  Jacobus H. van Lint,et al.  Equidistant codes with distance 12 , 1977, Discret. Math..

[17]  Alexander Vardy,et al.  Error-correcting codes in projective space , 2008, 2008 IEEE International Symposium on Information Theory.

[18]  Maximilien Gadouleau,et al.  Constant-Rank Codes and Their Connection to Constant-Dimension Codes , 2008, IEEE Transactions on Information Theory.

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

[20]  Heather Jordon,et al.  The maximum size of a partial 3-spread in a finite vector space over GF(2) , 2010, Des. Codes Cryptogr..

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

[22]  W. N. Hsieh,et al.  Intersection theorems for systems of finite vector spaces , 1975, Discret. Math..

[23]  Ute Rosenbaum,et al.  Projective Geometry: From Foundations to Applications , 1998 .

[24]  Balázs Patkós,et al.  Shadows and intersections in vector spaces , 2010, J. Comb. Theory, Ser. A.

[25]  Se June Hong,et al.  A General Class of Maximal Codes ror Computer Applications , 1972, IEEE Transactions on Computers.

[26]  David A. Drake,et al.  Partial t-spreads and group constructible (s,r,μ)-nets , 1979 .

[27]  Joachim Rosenthal,et al.  Decoding of Subspace Codes, a Problem of Schubert Calculus over Finite Fields , 2012, ArXiv.

[28]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

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