Diagnosing the Refuted Mismatched Decoding Converse for Binary-Input Channels
Abstract
We revisit the claimed converse theorem for mismatched decoding over binary-input discrete memoryless channels (Balakirsky, 1995) and the subsequent work refuting this converse via a numerical counter-example (Scarlett, Somekh-Baruch, Martinez, and Guillén i Fàbregas, 2015). A notable gap in the existing understanding is that no concrete flaw was identified in Balakirsky's analysis. In this paper, Balakirsky's proof is "diagnosed", and it is demonstrated that (i) his permutation lemma is flawed regardless of the correctness of his combinatorial approximation lemma; (ii) there are at least two specific incorrect steps regarding restricted codes and selector sequences, and these appear to be unlikely to admit a local repair; and (iii) the proposition in which these incorrect steps are used is false in general.
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BibTeX
@misc{scarlett2026diagnosing,
title = {{Diagnosing the Refuted Mismatched Decoding Converse for Binary-Input Channels}},
author = {Jonathan Scarlett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28109},
eprint = {2609.28109},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28109},
}
RIS
TY - UNPB
TI - Diagnosing the Refuted Mismatched Decoding Converse for Binary-Input Channels
AU - Jonathan Scarlett
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28109
UR - https://arxiv.org/abs/2609.28109
AB - We revisit the claimed converse theorem for mismatched decoding over binary-input discrete memoryless channels (Balakirsky, 1995) and the subsequent work refuting this converse via a numerical counter-example (Scarlett, Somekh-Baruch, Martinez, and Guillén i Fàbregas, 2015). A notable gap in the existing understanding is that no concrete flaw was identified in Balakirsky's analysis. In this paper, Balakirsky's proof is "diagnosed", and it is demonstrated that (i) his permutation lemma is flawed regardless of the correctness of his combinatorial approximation lemma; (ii) there are at least two specific incorrect steps regarding restricted codes and selector sequences, and these appear to be unlikely to admit a local repair; and (iii) the proposition in which these incorrect steps are used is false in general.
ER -