Exact random coding exponents for erasure decoding

Abstract

Abstract-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 moments 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.

Mismatched multi-letter successive decoding for the multiple-access channel

“…Our analysis is based on the method of type class enumeration (e.g. see [6], [9], [10]), and is perhaps most similar to that of Somekh-Baruch and Merhav [10]. We consider constantcomposition random coding, where for ν = 1, 2 we have…”

“…As noted by Grant et al [11], we can analyze the error probability of the second decoding step (see (3)) assuming that no error occurred on the first step (see (2)), while still using the unconditional statistics of (X 1 , X 2 , Y ). The subsequent analysis has been done in the study of maximum-metric decoding [3], [5], and the corresponding rate condition is precisely (10). In the remainder of this section, we focus on the first decoding step.…”

“…(17) The following lemma characterizes the behavior of N x1y ( P X1X2Y ) for fixed R 2 and P X1X2Y . The proof can be found in [6], [10], and is based on the fact that P (x 1 , X 2 , y) ∈ T n ( P X1X2Y )…”

“…[6], [10] Fix the pair (x 1 , y) ∈ T n ( P X1Y ), a constant δ > 0, and a type P X1X2Y ∈ S 1 (Q 2 , P X1Y ).…”

Codeword or noise? Exact random coding exponents for slotted asynchronism

“…The performance of this optimum detector/decoder is analyzed under the random coding regime of fixed composition codes, and the achievable trade-off between the three error exponents is given in full generality, that is, not merely in the margin where at least one of the exponents vanishes. We stress that our analysis technique, which is based on type class enumeration (see, e.g., [5], [7] and references therein), provides the exact random coding exponents, not just bounds. These relationships between the random coding exponents and the parameters α and β can, in principle, be inverted (in a certain domain) in order to find the assignments of α and β needed to satisfy given constraints on the exponents of the FA and the MD probabilities.…”