Rank Metric Decoder Architectures for Random Linear Network Coding With Error Control

While random linear network coding is a powerful tool for disseminating information in communication networks, it is highly susceptible to errors caused by various sources. Due to error propagation, errors greatly deteriorate the throughput of network coding and seriously undermine both reliability and security of data. Hence, error control for network coding is vital. Recently, constant-dimension codes (CDCs), especially Kötter-Kschischang (KK) codes, have been proposed for error control in random linear network coding. KK codes can also be constructed from Gabidulin codes, an important class of rank metric codes. Rank metric decoders have been recently proposed for both Gabidulin and KK codes, but they have high computational complexities. Furthermore, it is not clear whether such decoders are feasible and suitable for hardware implementations. In this paper, we reduce the complexities of rank metric decoders and propose novel decoder architectures for both codes. The synthesis results of our decoder architectures for Gabidulin and KK codes with limited error-correcting capabilities over small fields show that our architectures not only are affordable, but also achieve high throughput.

[1]  O. Ore.  On a special class of polynomials , 1933 .

[2]  O. Ore.  Contributions to the theory of finite fields , 1934 .

[3]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[4]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[5]  Herbert O. Burton.  Inversionless decoding of binary BCH codes , 1971, IEEE Trans. Inf. Theory.

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

[7]  Meghanad D. Wagh,et al.  A new structured design method for convolutions over finite fields, Part I , 1983, IEEE Trans. Inf. Theory.

[8]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

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

[10]  Ernst M. Gabidulin.  A Fast Matrix Decoding Algorithm for Rank-Error-Correcting Codes , 1991, Algebraic Coding.

[11]  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.

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

[13]  Naresh R. Shanbhag,et al.  High-speed architectures for Reed-Solomon decoders , 2001, IEEE Trans. Very Large Scale Integr. Syst..

[14]  M. Anwar Hasan,et al.  A New Construction of Massey-Omura Parallel Multiplier over GF(2m) , 2002, IEEE Trans. Computers.

[15]  Thomas Johansson,et al.  New Technique for Decoding Codes in the Rank Metric and Its Cryptography Applications , 2002, Probl. Inf. Transm..

[16]  Ning Cai,et al.  Network coding and error correction , 2002, Proceedings of the IEEE Information Theory Workshop.

[17]  Hanho Lee.  High-speed VLSI architecture for parallel Reed-Solomon decoder , 2003, IEEE Trans. Very Large Scale Integr. Syst..

[18]  K. Jain,et al.  Practical Network Coding , 2003 .

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

[20]  Bahram Honary,et al.  Reducible rank codes and their applications to cryptography , 2003, IEEE Trans. Inf. Theory.

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

[22]  Simon Plass,et al.  Error and Erasure Decoding of Rank-Codes with a Modified Berlekamp-Massey Algorithm , 2004 .

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

[24]  Shmuel Winograd.  Some bilinear forms whose multiplicative complexity depends on the field of constants , 2005, Mathematical systems theory.

[25]  K. K. Chi,et al.  Analysis of network error correction based on network coding , 2005 .

[26]  趙志宏.  Network Coding for Large Scale Content Distribution , 2005 .

[27]  James E. Stine,et al.  A framework for high-level synthesis of system on chip designs , 2005, 2005 IEEE International Conference on Microelectronic Systems Education (MSE'05).

[28]  Christos Gkantsidis,et al.  Network coding for large scale content distribution , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[29]  Muriel Médard,et al.  Codecast: a network-coding-based ad hoc multicast protocol , 2006, IEEE Wireless Communications.

[30]  Muriel Médard,et al.  Algebraic gossip: a network coding approach to optimal multiple rumor mongering , 2006, IEEE Transactions on Information Theory.

[31]  Andrey Bogdanov,et al.  A Parallel Hardware Architecture for fast Gaussian Elimination over GF(2) , 2006, 2006 14th Annual IEEE Symposium on Field-Programmable Custom Computing Machines.

[32]  Baochun Li,et al.  How Practical is Network Coding? , 2006, 200614th IEEE International Workshop on Quality of Service.

[33]  Tracey Ho,et al.  A Random Linear Network Coding Approach to Multicast , 2006, IEEE Transactions on Information Theory.

[34]  Paul D. Franzon,et al.  FreePDK: An Open-Source Variation-Aware Design Kit , 2007, 2007 IEEE International Conference on Microelectronic Systems Education (MSE'07).

[35]  F. Kschischang,et al.  A Rank-Metric Approach to Error Control in Random Network Coding , 2007, 2007 IEEE Information Theory Workshop on Information Theory for Wireless Networks.

[36]  Christina Fragouli,et al.  Wireless Network Coding: Opportunities & Challenges , 2007, MILCOM 2007 - IEEE Military Communications Conference.

[37]  Muriel Médard,et al.  XORs in the air: practical wireless network coding , 2008, TNET.

[38]  Maximilien Gadouleau,et al.  Complexity of decoding Gabidulin codes , 2008, 2008 42nd Annual Conference on Information Sciences and Systems.

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

[40]  Maximilien Gadouleau,et al.  On the Decoder Error Probability of Bounded Rank-Distance Decoders for Maximum RankDistance Codes , 2006, IEEE Transactions on Information Theory.

[41]  Ernst M. Gabidulin,et al.  Error and erasure correcting algorithms for rank codes , 2008, Des. Codes Cryptogr..

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

[43]  Ron M. Roth,et al.  Probabilistic algorithm for finding roots of linearized polynomials , 2008, Des. Codes Cryptogr..

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

[45]  Maximilien Gadouleau,et al.  Decoder error probability of bounded distance decoders for constant-dimension codes , 2009, 2009 IEEE International Symposium on Information Theory.

[46]  Zhiyuan Yan,et al.  On the decoder error probability of bounded rank distance decoders for rank metric codes , 2009, 2009 IEEE Information Theory Workshop.

[47]  Frank R. Kschischang,et al.  Fast encoding and decoding of Gabidulin codes , 2009, 2009 IEEE International Symposium on Information Theory.

[48]  Ning Chen,et al.  Cyclotomic FFTs With Reduced Additive Complexities Based on a Novel Common Subexpression Elimination Algorithm , 2007, IEEE Transactions on Signal Processing.

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