Exact Random Coding Exponents for Erasure Decoding

Random coding of channel decoding with an erasure option is studied. By analyzing the large deviations behavior of the code ensemble, we obtain exact single-letter formulas for the error exponents in lieu of Forney's lower bounds. The analysis technique we use is based on an enhancement and specialization of tools for assessing the statistical properties of certain distance enumerators. We specialize our results to the setup of the binary symmetric channel case with uniform random coding distribution and derive an explicit expression for the error exponent which, unlike Forney's bounds, does not involve optimization over two parameters. We also establish the fact that for this setup, the difference between the exact error exponent corresponding to the probability of undetected decoding error and the exponent corresponding to the erasure event is equal to the threshold parameter. Numerical calculations indicate that for this setup, as well as for a Z-channel, Forney's bound coincides with the exact random coding exponent.

[1]  S. Stenholm.  Information, Physics and Computation, by Marc Mézard and Andrea Montanari , 2010 .

[2]  Hesham El Gamal,et al.  On the Error Exponents of ARQ Channels With Deadlines , 2006, IEEE Transactions on Information Theory.

[3]  Neri Merhav.  The Generalized Random Energy Model and its Application to the Statistical Physics of Ensembles of Hierarchical Codes , 2009, IEEE Transactions on Information Theory.

[4]  Ning Cai,et al.  Erasure, list and detection zero-error capacities for low noise and a relation to identification , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

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

[6]  Neri Merhav,et al.  Error Exponents of Erasure/List Decoding Revisited Via Moments of Distance Enumerators , 2007, IEEE Transactions on Information Theory.

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

[8]  Neri Merhav,et al.  Exact Random Coding Exponents for Erasure Decoding , 2011, IEEE Trans. Inf. Theory.

[9]  Neri Merhav,et al.  Error Exponents of Optimum Decoding for the Interference Channel , 2010, IEEE Transactions on Information Theory.

[10]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[11]  Neri Merhav,et al.  Error Exponents for Broadcast Channels With Degraded Message Sets , 2011, IEEE Transactions on Information Theory.

[12]  Imre Csiszár,et al.  Information Theory - Coding Theorems for Discrete Memoryless Systems, Second Edition , 2011 .

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

[14]  Neri Merhav.  Relations Between Random Coding Exponents and the Statistical Physics of Random Codes , 2008, IEEE Transactions on Information Theory.

[15]  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.

[16]  Neri Merhav,et al.  Achievable Error Exponents for the Private Fingerprinting Game , 2007, IEEE Transactions on Information Theory.

[17]  Emre Telatar,et al.  Multi-access communications with decision feedback decoding , 1992 .

[18]  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.

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