Decoding interleaved Reed-Solomon codes over noisy channels

We consider error correction over the Non-Binary Symmetric Channel (NBSC) which is a natural probabilistic extension of the Binary Symmetric Channel (BSC). We propose a new decoding algorithm for interleaved Reed-Solomon codes that attempts to correct all ''interleaved'' codewords simultaneously. In particular, interleaved encoding gives rise to multi-dimensional curves and more specifically to a variation of the Polynomial Reconstruction Problem, which we call Simultaneous Polynomial Reconstruction. We present and analyze a novel probabilistic algorithm that solves this problem. Our construction yields a decoding algorithm for interleaved RS codes that allows efficient transmission arbitrarily close to the channel capacity in the NBSC model.

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

[2]  Paul C. van Oorschot,et al.  An Introduction to Error Correcting Codes with Applications , 1989 .

[3]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[4]  Aggelos Kiayias,et al.  Secure Games with Polynomial Expressions , 2001, ICALP.

[5]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometric codes , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[6]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[7]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[8]  Aggelos Kiayias,et al.  Decoding of Interleaved Reed Solomon Codes over Noisy Data , 2003, ICALP.

[9]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[10]  Tomasz Radzik,et al.  Automata, Languages and Programming, 36th International Colloquium , ICALP 2009 , 2009, ICALP 2009.

[11]  Michael O. Rabin,et al.  Efficient dispersal of information for security, load balancing, and fault tolerance , 1989, JACM.

[12]  Madhu Sudan,et al.  Decoding of Reed Solomon Codes beyond the Error-Correction Bound , 1997, J. Complex..

[13]  G. David Forney,et al.  Concatenated codes , 2009, Scholarpedia.

[14]  Elwyn R. Berlekamp Bounded distance+1 soft-decision Reed-Solomon decoding , 1996, IEEE Trans. Inf. Theory.

[15]  Madhu Sudan,et al.  Reconstructing curves in three (and higher) dimensional space from noisy data , 2003, STOC '03.

[16]  Hugo Krawczyk Distributed fingerprints and secure information dispersal , 1993, PODC '93.