Binary code rate bounds via classical--quantum channels

2026-08-10

AI Summary

This work unifies and improves classical asymptotic rate–distance bounds for binary codes by reframing the problem as one of channel design: minimizing channel capacity subject to a constraint on the posterior bit error probability. Leveraging classical–quantum channels and the Pretty Good Measurement (PGM) criterion, the authors construct an analytical framework that, for the first time, derives the Plotkin bound, the Elias–Bassalygo bound, and both MRRW bounds within a single coherent approach. Central to this framework is the introduction of a novel quantum channel model—namely, the Mixed Qubit Channel (MQC) and its masked variant. The proposed method yields strictly tighter asymptotic bounds than the classical MRRW bound over the relative distance interval (0, 1/2), thereby achieving a significant improvement over existing results.

Abstract

We derive the four principal asymptotic rate-distance tradeoffs for binary codes---Plotkin, Elias--Bassalygo, and the two McEliece--Rodemich--Rumsey--Welch (MRRW) bounds---from one theorem, the ``pretty good criterion.'' If the bit error rate under the pretty good measurement (PGM)---the quantum analog of posterior sampling---of a binary-input output-symmetric classical--quantum (cq) channel lies below $δ$, then every length-$n$ binary code, linear or nonlinear, of relative distance $δ$ has rate at most the channel's capacity, up to an $O(n^{-1/2})$ correction. Rate--distance bounds thereby reduce to a channel design problem, wherein the task is to minimize channel capacity subject to the posterior bit error rate constraint. Via the pretty good criterion, the binary erasure channel (BEC) yields Plotkin, the binary symmetric channel (BSC) yields Elias--Bassalygo, the pure-state channel (PSC) yields the first MRRW bound, and a masked PSC yields the second MRRW bound exactly. This framework is then instantiated with new channels to improve upon the MRRW bounds. Specifically, the mixed-qubit channel (MQC), a mixed-state version of PSC, strictly improves the first MRRW bound at every $0 < δ< \frac{1}{2}$, while the masked mixed-qubit channel (2MQC) strictly improves the second MRRW bound throughout the same interval.

Problem

• binary codes
• rate-distance bounds
• MRRW bounds
• classical-quantum channels
• asymptotic tradeoffs

Innovation

• classical-quantum channels
• pretty good measurement
• rate-distance bounds
• mixed-qubit channel
• MRRW bounds