Error Exponents for Randomised List Decoding
Henrique K. Miyamoto, Sheng Yang
TL;DR
The paper analyzes randomised list decoding over discrete memoryless channels, deriving error-exponent characterisations for two regimes: fixed list size ${L}$ and exponential list size ${L(n)=e^{n\lambda}}$. For fixed ${L}$, it provides an ensemble-tight exponent $E_1(R,Q,L)$ under mismatched metrics, and shows that for matched (and MMI) metrics the exponent collapses to the ordinary random-coding exponent $E_{\mathrm{r}}(R,Q)$, implying no improvement from randomised lists in this regime. In the exponential-list regime, it proves a non-trivial lower bound $E_2(R,Q,\lambda)$, which matches $E_{\mathrm{r}}(R-\lambda)$ for matched and MMI metrics at high rates (above $R_0+\lambda$), and shows that rates up to $R+\lambda$ are achievable with randomised list decoding. The results also extend to a universal, channel-independent metric (the randomised analogue of MMI), preserving the exponent gains in the universal setting. Overall, the work clarifies when randomised list decoding improves error exponents and how the gains depend on list size growth and metric choice.
Abstract
This paper studies random-coding error exponents of randomised list decoding, in which the decoder randomly selects $L$ messages with probabilities proportional to the decoding metric of the codewords. The exponents (or bounds) are given for mismatched, and then particularised to matched and universal decoding metrics. Two regimes are studied: for fixed list size, we derive an ensemble-tight random-coding error exponent, and show that, for the matched metric, it does not improve the error exponent of ordinary decoding. For list sizes growing exponentially with the block-length, we provide a non-trivial lower bound to the error exponent that is tight at high rates under the matched metric.