The Guesswork of Ordered Statistics Decoding: Guesswork Complexity and Decoder Design

TL;DR

This work develops tight non-asymptotic bounds on the guesswork required by ordered statistics decoding (OSD) over binary AWGN channels, by modeling the decoding process as ordered test-error-pattern exploration and applying Hölder-type bounds with Hamming-shell partitions. It extends these results to ordered statistics via conditional independence (BI-OSC) and derives both exact and practical approximations (including modified Bessel function forms and binomial/Poisson tails) for the average and higher moments of guesswork, distinguishing order-$k$ and order-$m$ OSD behavior. A key finding is a saturation threshold $m_s=\lceil k\sqrt{p_e}\rceil$ beyond which increasing the decoding order ceases to raise the average guesswork, providing a principled guide for selecting OSD parameters and enabling efficient HARQ designs with complexity control. The results yield actionable insights for OSD algorithm design, including CRC-based correct-word identification and a complexity-cutoff criterion (CCC) to bound worst-case guesswork while maintaining near-MLD error performance, thereby supporting practical deployment of OSD in short-blocklength, high-reliability settings.

Abstract

This paper investigates guesswork over ordered statistics and formulates the achievable guesswork complexity of ordered statistics decoding (OSD) in binary additive white Gaussian noise (AWGN) channels. The achievable guesswork complexity is defined as the number of test error patterns (TEPs) processed by OSD immediately upon finding the correct codeword estimate. The paper first develops a new upper bound for guesswork over independent sequences by partitioning them into Hamming shells and applying Hölder's inequality. This upper bound is then extended to ordered statistics, by constructing the conditionally independent sequences within the ordered statistics sequences. Next, we apply these bounds to characterize the statistical moments of the OSD guesswork complexity. We show that the achievable guesswork complexity of OSD at maximum decoding order can be accurately approximated by the modified Bessel function, which increases exponentially with code dimension. We also identify a guesswork complexity saturation threshold, where increasing the OSD decoding order beyond this threshold improves error performance without further raising the achievable guesswork complexity. Finally, the paper presents insights on applying these findings to enhance the design of OSD decoders.

Figures (25)

• Figure 1: The average number of guesses at different blocklength $n$ with binary symmetric channel with error probability $0.05$.
• Figure 2: The average number of guesses for various block lengths $n$ in a BI-AWGN channel for the pair $(X^n,Y^n)$ at SNR = 0 dB. The suboptimal guesswork is performed in ascending order of the Hamming distance from the most likely $X^n$ sequence given $Y^n$.
• Figure 3: The performance of the proposed Hamming subset bound with various values of parameter $q$ for blocklengths $n = 20$ in a BI-AWGN channel at SNR = 0 dB.
• Figure 4: The average number of guesses at different blocklength $n$ in a BI-OSC derived from an AWGN channel at SNR = 3 dB. The guesswork focuses on the $[n/4, n/2]$ ordered channel outputs range. The suboptimal guesswork is performed in ascending order of the Hamming distance to the most likely sequence $\widetilde{X}_a^b$ when $\widetilde{Y}_a^b$ is given.
• Figure 5: The performance of Theorem \ref{['The::Newbound:XnYn:ordered']} for the average number of guesses with various values of parameter $q$ for $n = 64$ in a BI-OSC derived from an AWGN channel at SNR = 3 dB.

Theorems & Definitions (19)

• Theorem 1: Arikan's lower bound arikan1996inequality
• Theorem 2: Markov Source malone2004guesswork
• Theorem 3: Existence of Rényi entropy rate hanawal2010guessing
• Theorem 4: LDP of guesswork christiansen2012guesswork
• Lemma 1
• Theorem 5: Hamming subset bound
• Corollary 1
• Lemma 2
• Corollary 2
• Theorem 6