Guessing Decoding of Short Blocklength Codes

TL;DR

This work addresses decoding for short-blocklength codes in ultra-reliable, low-latency regimes by unifying two universal guessing-decoding families, GRAND and GCD. It develops algorithmic implementations, proves $ML$ optimality under appropriate stopping, and provides saddle-point analyses for average query counts, validating results with simulations. The study quantifies the performance gap under finite search budgets, compares key metrics, and demonstrates cross-pollination between GRAND and GCD—along with practical hardware considerations—across high- to low-rate regimes. The findings show GRAND excels at high rates while GCD dominates at low rates, with ORB-based and DAI enhancements enabling near-ML performance with manageable complexity, guiding deployment for next-generation short-blocklength links.

Abstract

Future beyond-5G and 6G systems demand ultra-reliable, low-latency communication with short blocklengths, motivating the development of universal decoding algorithms. Guessing decoding, which infers the noise or codeword candidate in order of decreasing (exact or approximate) likelihood, offers a universal framework applicable to short codes. In this paper, we present a unified treatment of two prominent recent families of guessing decoding: guessing random additive noise decoding (GRAND) and guessing codeword decoding (GCD). For each, we (i) present algorithmic implementations and ordering strategies; (ii) prove maximum-likelihood (ML) optimality under appropriate stopping criteria; (iii) derive saddle-point approximations for the average number of queries; and (iv) validate theoretical predictions with simulations. We further analyze the performance degradation due to limited search budgets relative to ML performance, compare key metrics (worst-case and average complexity, hardware considerations), and highlight how advances in one approach transfer naturally to the other. Our results clarify the operating regimes where GRAND and GCD demonstrate superior performance. This work provides both theoretical insights and practical guidelines for deploying universal guessing decoders in next-generation short-blocklength communications.

Figures (7)

• Figure 1: Milestones in the development of guessing decoding for short codes. The timeline highlights key algorithmic advances from Chase decoding to modern GRAND and GCD variants.
• Figure 2: Practical operating regimes for three classes of guessing-based decoders over BPSK-AWGN channels. GRAND is best suited for high-rate codes, OSD performs well in the medium-rate regime due to GE, and GCD offers strong performance for low-rate codes.
• Figure 3: The performance of the random linear code $\mathscr{C}[128,106]$ under SGRAND duffy2019capacity and ORBGRAND duffy2022ordered, where $\ell_{\max} = 5\times 10^6$.
• Figure 4: The performance of the eBCH code $\mathscr{C}[128,22]$ under GCD with trivial and DAI stopping criteria, where $\ell_{\max} = 5\times 10^6$.
• Figure 5: The upper bound on the performance gap to the ML decoding for the linear block code $\mathscr{C}[128,106]$ over BPSK-AWGN channels, where we consider GRAND duffy2019capacity and GCD Zheng2024IT.

Theorems & Definitions (13)

• Example 1: Comparison of Ordering Strategies
• Definition 1: Membership-Check Stopping Criterion
• Proposition 1
• Example 2: GRAND with Soft-Weight Order versus ORB Order
• Definition 2: Trivial Stopping Criterion
• Proposition 2
• Definition 3: DAI Stopping Criterion
• Example 3: Trivial versus DAI Stopping Criterion
• Example 4: Upper Bound on the Gap to ML Decoding
• Example 5: Minimum Required $\widetilde{\ell}_{\max}$ versus Code Rate