On the Decoder Error Probability of Bounded Rank-Distance Decoders for Maximum RankDistance Codes

In this correspondence, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable codes. Using these properties, we show that, for MRD codes with error correction capability , the decoder error probability of bounded rank distance decoders decreases exponentially with based on the assumption that all errors with the same rank are equally likely.

[1]  Kar-Ming Cheung More on the decoder error probability for Reed-Solomon codes , 1989, IEEE Trans. Inf. Theory.

[2]  L. Tolhuizen A Universal Upper Bound on the Miscorrection Probability with Bounded Distance Decoding for a Code Used on an Error-Value Symmetric Channel , 1993 .

[3]  Robert J. McEliece,et al.  On the decoder error probability for Reed-Solomon codes , 1986, IEEE Trans. Inf. Theory.

[4]  Se June Hong,et al.  Optimal Rectangular Code for High Density Magnetic Tapes , 1974, IBM J. Res. Dev..

[5]  Moshe Schwartz,et al.  Two-dimensional cluster-correcting codes , 2005, IEEE Transactions on Information Theory.

[6]  W. F. Mikhail,et al.  The Reliability of Memory with Single-Error Correction , 1982, IEEE Transactions on Computers.

[7]  Jehoshua Bruck,et al.  MDS Array Codes for Correcting Criss-Cross Errors , 1997 .

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

[9]  Stanislaw Budkowski,et al.  A Double Track Error-Correction Code for Magnetic Tape , 1976, IEEE Transactions on Computers.

[10]  HIDEKI IMAI,et al.  Two-dimensional Fire codes , 1973, IEEE Trans. Inf. Theory.

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

[12]  Martin Bossert,et al.  Maximum rank distance codes as space-time codes , 2003, IEEE Trans. Inf. Theory.

[13]  M. Anthony,et al.  Advanced linear algebra , 2006 .

[14]  H. Keng,et al.  A THEOREM ON MATRICES OVER A SFIELD AND ITS APPLICATIONS , 1951 .

[15]  Pierre Loidreau Decoding rank errors beyond the error-correcting capability , 2006 .

[16]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

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

[18]  Jim Hefferon,et al.  Linear Algebra , 2012 .

[19]  M. Blaum,et al.  Coding protection for magnetic tapes: A generalization of the Patel - Hong code , 1985, IEEE Trans. Inf. Theory.

[20]  Hideki Imai,et al.  A Theory of Two-Dimensional Cyclic Codes , 1977, Inf. Control..

[21]  Pierre Loidreau,et al.  A Welch-Berlekamp Like Algorithm for Decoding Gabidulin Codes , 2005, WCC.

[22]  Ernst M. Gabidulin,et al.  The new construction of rank codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[23]  R. Blahut Theory and practice of error control codes , 1983 .

[24]  Ernst M. Gabidulin,et al.  Ideals over a Non-Commutative Ring and thier Applications in Cryptology , 1991, EUROCRYPT.

[25]  Richard E. Blahut,et al.  The Capacity and Coding Gain of Certain Checkerboard Codes , 1998, IEEE Trans. Inf. Theory.

[26]  Patrick Guy Farrell,et al.  A survey of array error control codes , 1992, Eur. Trans. Telecommun..

[27]  M. Blaum,et al.  Array codes for cluster-error correction , 1994 .

[28]  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.

[29]  Martin Bossert,et al.  Array Codes Correcting a Two-Dimensional Cluster of Errors , 1998, IEEE Trans. Inf. Theory.

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

[31]  Ron M. Roth,et al.  Probabilistic crisscross error correction , 1997, IEEE Trans. Inf. Theory.

[32]  Simon Plass,et al.  Fast decoding of rank-codes with rank errors and column erasures , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..