Rank distribution of Delsarte codes

In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for Delsarte rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to Gabidulin codes.

[1]  Frank R. Kschischang,et al.  Universal Secure Network Coding via Rank-Metric Codes , 2008, IEEE Transactions on Information Theory.

[2]  Heide Gluesing-Luerssen,et al.  Fourier-reflexive partitions and MacWilliams identities for additive codes , 2013, Des. Codes Cryptogr..

[3]  Rudolf Ahlswede,et al.  Network information flow , 2000, IEEE Trans. Inf. Theory.

[4]  Muriel Medard,et al.  Network coding : fundamentals and applications , 2012 .

[5]  Frank R. Kschischang,et al.  On Metrics for Error Correction in Network Coding , 2008, IEEE Transactions on Information Theory.

[6]  Maximilien Gadouleau,et al.  MacWilliams Identity for the Rank Metric , 2007, 2007 IEEE International Symposium on Information Theory.

[7]  Frédérique E. Oggier,et al.  On the existence of generalized rank weights , 2012, 2012 International Symposium on Information Theory and its Applications.

[8]  Marta Giorgetti,et al.  Galois invariance, trace codes and subfield subcodes , 2010, Finite Fields Their Appl..

[9]  T. Ho,et al.  On Linear Network Coding , 2010 .

[10]  Alfred Wassermann,et al.  Algebraic structures of MRD codes , 2015, Adv. Math. Commun..

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

[12]  Tomohiko Uyematsu,et al.  Relative Generalized Rank Weight of Linear Codes and Its Applications to Network Coding , 2013, IEEE Transactions on Information Theory.

[13]  Marcus Greferath,et al.  MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings , 2013, J. Comb. Theory, Ser. A.

[14]  Maximilien Gadouleau,et al.  MacWilliams Identity for Codes with the Rank Metric , 2007, EURASIP J. Wirel. Commun. Netw..

[15]  John Sheekey,et al.  A new family of linear maximum rank distance codes , 2015, Adv. Math. Commun..

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

[17]  Mario A. de Boer,et al.  Almost MDS codes , 1996, Des. Codes Cryptogr..

[18]  I. G. Núñez,et al.  Generalized Hamming Weights for Linear Codes , 2001 .

[19]  Alberto Ravagnani,et al.  Generalized weights: an anticode approach , 2014, ArXiv.

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

[21]  G. Andrews.  The Theory of Partitions: Frontmatter , 1976 .

[22]  Wolfgang Willems,et al.  Codes of Small Defect , 1997, Des. Codes Cryptogr..