Tables of subspace codes

One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least $d$ over $\mathbb{F}_q^n$, where the dimensions of the codewords, which are vector spaces, are contained in $K\subseteq\{0,1,\dots,n\}$. In the special case of $K=\{k\}$ one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{this http URL}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.

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

[2]  Sascha Kurz,et al.  Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound , 2017, ICMCTA.

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

[4]  Sascha Kurz,et al.  Coset Construction for Subspace Codes , 2015, IEEE Transactions on Information Theory.

[5]  Joachim Rosenthal,et al.  Cyclic Orbit Codes , 2011, IEEE Transactions on Information Theory.

[6]  A. Iranmanesh,et al.  Cyclic Orbit Codes with the Normalizer of a Singer Subgroup , 2015 .

[7]  Antonio Cossidente,et al.  Geometrical Aspects of Subspace Codes , 2018 .

[8]  Shu-Tao Xia,et al.  A Graham-Sloane Type Construction of Constant Dimension Codes , 2008, 2008 Fourth Workshop on Network Coding, Theory and Applications.

[9]  Alfred Wassermann,et al.  The order of the automorphism group of a binary $${\varvec{q}}$$q-analog of the Fano plane is at most two , 2018, Des. Codes Cryptogr..

[10]  Johannes André,et al.  Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe , 1954 .

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

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

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

[14]  Joachim Rosenthal,et al.  A Complete Characterization of Irreducible Cyclic Orbit Codes , 2011, ArXiv.

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

[16]  Sascha Kurz.  Improved upper bounds for partial spreads , 2017, Des. Codes Cryptogr..

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

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

[19]  Alfred Wassermann,et al.  Projective divisible binary codes , 2017 .

[20]  Anirban Ghatak.  Optimal Binary (5, 3) Projective Space Codes from Maximal Partial Spreads , 2017, ArXiv.

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

[22]  Ismael Gutierrez,et al.  Some constructions of cyclic and quasi-cyclic subspaces codes , 2015 .

[23]  Ferruh Özbudak,et al.  Cyclic subspace codes via subspace polynomials , 2017, Des. Codes Cryptogr..

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

[25]  Yong Jiang,et al.  On the Linear Programming Bounds for Constant Dimension Codes , 2011, 2011 International Symposium on Networking Coding.

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

[27]  Joachim Rosenthal,et al.  A complete characterization of irreducible cyclic orbit codes and their Plücker embedding , 2011, Des. Codes Cryptogr..

[28]  R. Ahlswede,et al.  On error control codes for random network coding , 2009, 2009 Workshop on Network Coding, Theory, and Applications.

[29]  Heide Gluesing-Luerssen,et al.  Cyclic orbit codes and stabilizer subfields , 2015, Adv. Math. Commun..

[30]  Tuvi Etzion,et al.  Galois geometries and coding theory , 2016, Des. Codes Cryptogr..

[31]  Patric R. J. Östergård,et al.  New Lower Bounds for Binary Constant-Dimension Subspace Codes , 2018, Exp. Math..

[32]  Sascha Kurz.  Heden's bound on the tail of a vector space partition , 2018, Discret. Math..

[33]  Olof Heden.  On the length of the tail of a vector space partition , 2009, Discret. Math..

[34]  Vojtech Rödl,et al.  Near Perfect Coverings in Graphs and Hypergraphs , 1985, Eur. J. Comb..

[35]  A. Trautmann,et al.  Constructions, decoding and automorphisms of subspace codes , 2013 .

[36]  Vitaly Skachek,et al.  Recursive Code Construction for Random Networks , 2008, IEEE Transactions on Information Theory.

[37]  Joachim Rosenthal,et al.  Correction to “Cyclic Orbit Codes” , 2017, IEEE Transactions on Information Theory.

[38]  Tao Feng,et al.  Several classes of optimal Ferrers diagram rank-metric codes , 2018, Linear Algebra and its Applications.

[39]  Natalia Silberstein,et al.  Subspace Codes Based on Graph Matchings, Ferrers Diagrams, and Pending Blocks , 2014, IEEE Transactions on Information Theory.

[40]  Sascha Kurz,et al.  An upper bound for binary subspace codes of length 8, constant dimension 4 and minimum distance 6 , 2017 .

[41]  P. Delsarte.  AN ALGEBRAIC APPROACH TO THE ASSOCIATION SCHEMES OF CODING THEORY , 2011 .

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

[43]  Joachim Rosenthal,et al.  On conjugacy classes of subgroups of the general linear group and cyclic orbit codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[44]  Michael Braun,et al.  q‐Analogs of Packing Designs , 2012, 1212.4614.

[45]  Sascha Kurz,et al.  Partial spreads and vector space partitions , 2016, 1611.06328.

[46]  Joachim Rosenthal,et al.  Orbit codes — A new concept in the area of network coding , 2010, 2010 IEEE Information Theory Workshop.

[47]  Papa Sissokho,et al.  The maximum size of a partial spread in a finite projective space , 2016, J. Comb. Theory, Ser. A.

[48]  Natalia Silberstein,et al.  New lower bounds for constant dimension codes , 2013, 2013 IEEE International Symposium on Information Theory.

[49]  J. Macwilliams.  A theorem on the distribution of weights in a systematic code , 1963 .

[50]  Selmer M. Johnson.  A new upper bound for error-correcting codes , 1962, IRE Trans. Inf. Theory.

[51]  P. Delsarte.  Hahn Polynomials, Discrete Harmonics, and t-Designs , 1978 .

[52]  Antonio Cossidente,et al.  Veronese subspace codes , 2016, Des. Codes Cryptogr..

[53]  Sascha Kurz,et al.  Binary Subspace Codes in Small Ambient Spaces , 2018, Adv. Math. Commun..

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

[55]  Thomas Honold,et al.  On putative q-analogues of the Fano plane and related combinatorial structures , 2015, 1504.06688.

[56]  Sascha Kurz,et al.  Constructions and bounds for mixed-dimension subspace codes , 2015, Adv. Math. Commun..

[57]  Christine Bachoc,et al.  Bounds for projective codes from semidefinite programming , 2012, Adv. Math. Commun..

[58]  Rahim Tafazolli,et al.  Network Coding Theory: A Survey , 2013, IEEE Communications Surveys & Tutorials.

[59]  Joachim Rosenthal,et al.  Constructions of Constant Dimension Codes , 2018 .

[60]  Sascha Kurz.  Packing vector spaces into vector spaces , 2017, Australas. J Comb..

[61]  Sascha Kurz,et al.  Classification of large partial plane spreads in $${{\,\mathrm{PG}\,}}(6,2)$$PG(6,2) and related combinatorial objects , 2018, Journal of Geometry.

[62]  Ron M. Roth,et al.  Construction of Sidon Spaces With Applications to Coding , 2017, IEEE Transactions on Information Theory.

[63]  Tuvi Etzion,et al.  The Asymptotic Behavior of Grassmannian Codes , 2011, IEEE Transactions on Information Theory.

[64]  Ivan Molina Naizir,et al.  On quasi-cyclic subspace codes , 2016 .

[65]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[66]  Heide Gluesing-Luerssen,et al.  Construction of subspace codes through linkage , 2015, Adv. Math. Commun..

[67]  Eli Ben-Sasson,et al.  Subspace Polynomials and Cyclic Subspace Codes , 2014, IEEE Transactions on Information Theory.

[68]  Sascha Kurz,et al.  An improvement of the Johnson bound for subspace codes , 2017, ArXiv.

[69]  Papa Sissokho,et al.  The maximum size of a partial spread II: Upper bounds , 2016, Discret. Math..

[70]  Sascha Kurz.  Upper bounds for partial spreads , 2017 .

[71]  Svetlana Topalova,et al.  Line spreads of PG(5, 2) , 2009 .

[72]  Joan-Josep Climent,et al.  A Construction of Orbit Codes , 2017, ICMCTA.

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

[74]  Antonio Cossidente,et al.  Subspace Codes in PG(2N − 1, Q) , 2014, Combinatorica.

[75]  Thomas Honold,et al.  The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes , 2016, ArXiv.

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

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

[78]  Ernst Gabidulin,et al.  On cardinality of network subspace codes 1 , 2014 .

[79]  Alberto Ravagnani,et al.  Optimal Ferrers Diagram Rank-Metric Codes , 2014, IEEE Transactions on Information Theory.

[80]  Sascha Kurz,et al.  Generalized vector space partitions , 2019, Australas. J Comb..

[81]  Qing Xiang,et al.  New bounds for partial spreads of H(2d - 1, q2) and partial ovoids of the Ree-Tits octagon , 2018, J. Comb. Theory, Ser. A.

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

[83]  Ernst M. Gabidulin,et al.  Rank subcodes in multicomponent network coding , 2013, Probl. Inf. Transm..

[84]  Thomas Honold,et al.  Poster: A new approach to the Main Problem of Subspace Coding , 2014, 9th International Conference on Communications and Networking in China.