The Error Probability of Generalized Perfect Codes via the Meta-Converse
Gonzalo Vazquez-Vilar, Albert Guill\'en i F\`abregas, Sergio Verd\'u
TL;DR
This paper generalizes the concept of perfect and quasi-perfect codes for symmetric channels, showing their error probabilities match the meta-converse bound and extending to source coding and compression.
Contribution
It introduces a new generalized definition of perfect codes for symmetric channels that includes MDS codes and links their error probability to the meta-converse bound.
Findings
Error probability of these codes equals the meta-converse lower bound.
The definition extends to almost-lossless source-channel coding.
The framework encompasses maximum distance separable codes.
Abstract
We introduce a definition of perfect and quasi-perfect codes for symmetric channels parametrized by an auxiliary output distribution. This notion generalizes previous definitions of perfect and quasi-perfect codes and encompasses maximum distance separable codes. The error probability of these codes, whenever they exist, is shown to coincide with the estimate provided by the meta-converse lower bound. We illustrate how the proposed definition naturally extends to cover almost-lossless source-channel coding and lossy compression.
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