Universal Decoding With an Erasure Option

Motivated by applications of rateless coding, decision feedback, and automatic repeat request (ARQ), we study the problem of universal decoding for unknown channels in the presence of an erasure option. Specifically, we harness the competitive minimax methodology developed in earlier studies, in order to derive a universal version of Forney's classical erasure/list decoder, which in the erasure case, optimally trades off between the probability of erasure and the probability of undetected error. The proposed universal erasure decoder guarantees universal achievability of a certain fraction xi of the optimum error exponents of these probabilities. A single-letter expression for xi, which depends solely on the coding rate and the Neyman-Pearson threshold, is provided. The example of the binary symmetric channel is studied in full detail, and some conclusions are drawn.

[1]  M. Feder,et al.  Static broadcasting , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[2]  G. David Forney,et al.  Exponential error bounds for erasure, list, and decision feedback schemes , 1968, IEEE Trans. Inf. Theory.

[3]  Meir Feder,et al.  Universal decoders for channels with memory , 1996 .

[4]  Neri Merhav,et al.  Minimax Universal Decoding With an Erasure Option , 2007, IEEE Transactions on Information Theory.

[5]  Frank R. Kschischang,et al.  Rateless Coding for Non-Ergodic Channels with Decoder Channel State Information , 2006 .

[6]  Brendan J. Frey,et al.  Efficient variable length channel coding for unknown DMCs , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[7]  Krishna R. Narayanan,et al.  Multilevel Coding for Channels with Non-uniform Inputs and Rateless Transmission over the BSC , 2006, 2006 IEEE International Symposium on Information Theory.

[8]  J. Massey.  CAUSALITY, FEEDBACK AND DIRECTED INFORMATION , 1990 .

[9]  Rudolf Ahlswede,et al.  Erasure, list, and detection zero-error capacities for low noise and a relation to identification , 1996, IEEE Trans. Inf. Theory.

[10]  Takeshi Hashimoto,et al.  Composite Scheme LR + Th for Decoding with Erasures and Its Effective Equivalence to Forney's Rule , 1999, IEEE Trans. Inf. Theory.

[11]  Michael Mitzenmacher,et al.  A digital fountain approach to asynchronous reliable multicast , 2002, IEEE J. Sel. Areas Commun..

[12]  Daniel A. Spielman,et al.  Efficient erasure correcting codes , 2001, IEEE Trans. Inf. Theory.

[13]  Nadav Shulman,et al.  Communication over an unknown channel via common broadcasting , 2003 .

[14]  Takeshi Hashimoto,et al.  Performance of explicit error detection and threshold decision in decoding with erasures , 1997, IEEE Trans. Inf. Theory.

[15]  Jacob Ziv,et al.  Universal decoding for finite-state channels , 1985, IEEE Trans. Inf. Theory.

[16]  Neri Merhav,et al.  Universal composite hypothesis testing: A competitive minimax approach , 2002, IEEE Trans. Inf. Theory.

[17]  Michael Luby,et al.  A digital fountain approach to reliable distribution of bulk data , 1998, SIGCOMM '98.

[18]  Meir Feder,et al.  Universal Decoding for Channels with Memory , 1998, IEEE Trans. Inf. Theory.

[19]  Meir Feder,et al.  The uniform distribution as a universal prior , 2004, IEEE Transactions on Information Theory.

[20]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[21]  Gregory W. Wornell,et al.  Rateless space-time coding , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[22]  Andrew J. Viterbi,et al.  Error bounds for the white Gaussian and other very noisy memoryless channels with generalized decision regions , 1969, IEEE Trans. Inf. Theory.