A Dual-Domain Achievability of the Typical Error Exponent

Abstract

For random-coding ensembles with pairwiseindependent codewords, we show that the probability that the exponent of a given code from the ensemble being smaller than an upper bound on the typical random-coding exponent is vanishingly small. This upper bound is known to be tight for i.i.d. ensembles over the binary symmetric channel and for constant-composition codes over memoryless channels. Our result recovers these as special cases and remains valid for arbitrary alphabets and channel memory, as well as arbitrary ensembles with pairwise independent codewords.

Typical Random Coding Exponent for Finite-State Channels

“…In [15], we studied the probability of selecting a code C n from the ensemble with an error exponent (3) larger than a finite-length lower bound version of the TRC exponent (5). We showed that such probability converges to one as the codeword length n tends to infinity for channels with arbitrary alphabets or memory, with the only assumption of pairwise-independent codewords.…”

“…The first lower bound used in (7), meaningful for low rates and based on our previous work in [15], is given by…”

“…To sum up, if R ∞ (Q) > 0, the TRC exponent is infinite for all rates R < R ∞ (Q). The cases in which R ∞ (Q) = 0 and the analysis of the optimal λ for R = 0 can be derived in a similar fashion as done here and in [15], respectively, and are not reported here for a matter of space. As a final remark, we complement Lemma 2 by noticing that from the discussion above it follows that…”

Generalized Random Gilbert-Varshamov Codes: Typical Error Exponent and Concentration Properties

“…The TRC was shown to be universally achievable with a likelihood mutual-information decoder in [17]. For pairwise-independent ensembles and arbitrary channels, Cocco et al showed in [18] that the probability that a code in the ensemble has an exponent smaller than a lower bound on the TRC exponent is vanishingly small. Recently, Truong et al showed that, for DMCs, the error exponent of a ran-domly generated code with pairwise-independent codewords converges in probability to its expectation -the typical error exponent [19].…”

Concentration Properties of Random Codes

“…The TRC was shown to be universally achievable with the likelihood mutual information decoder in [14]. For pairwise-independent ensembles and arbitrary channels, Cocco et al showed in [15] that the probability that the exponent of a given code in the ensemble is smaller than a lower bound on the TRC exponent is vanishingly small.…”