Close the logarithmic-term gap for VLSF codes

Close the gap between the general VLSF achievability coefficient -1 and the converse coefficient -C/C_1 in the log N term of the VLSF fundamental limit for discrete memoryless channels with positive capacity C and finite largest pairwise conditional-output KullbackLeibler divergence C_1.

Background

Variable-length stop-feedback codes are a subclass of variable-length feedback codes in which the channel-input sequence is fixed by the message and common randomness and does not depend on past channel outputs; feedback only informs the encoder when decoding occurs. Consequently, any converse bound for variable-length feedback codes also applies to VLSF codes.

For general VLSF coding, the best achievability result cited in the paper has a second-order term with coefficient -1 multiplying log N, whereas the paper’s converse for discrete memoryless channels with C>0 and finite C_1 has coefficient -C/C_1. Since C<C_1 for every positive-capacity discrete memoryless channel, these coefficients differ, leaving the second-order VLSF fundamental limit unresolved.

References

To the best of our knowledge, the lower bound in~eq:intro-ppv remains the best general VLSF achievability bound, and its coefficient of $\log N$ is $-1$. Since $C<C_1$ for every DMC with $C\>0$, the achievability coefficient $-1$ differs from the converse coefficient $-\frac{C}{C_1}$. Closing this gap remains an open problem.

— A Tight Second-Order Converse Bound for Variable-Length Feedback Codes  (2609.11368 - Yavas, 10 Sep 2026) in Section 1, Introduction, paragraph beginning “In a variable-length stop-feedback (VLSF) code”