Using a Single-Parity-Check to Reduce the Guesswork of Guessing Codeword Decoding

TL;DR

The paper addresses the high guesswork of Guessing Codeword Decoding (GCD) by introducing SA-GCD, a joint decoding framework that leverages an SPC outer code within a concatenated binary linear code and a landslide-configured list-ORBGRAND to reorder information-theory guided guesses. By exploiting the even-code parity constraint of the SPC outer code, SA-GCD shifts low-cost guesses later in the order, reducing total guesswork by up to about a factor of 2 at lower SNRs without degrading BLER. The approach relies on a practical outer-inner code structure and configurable noise-pattern generation, enabling significant reductions in decoding complexity for soft-input decoding schemes. The results demonstrate tangible complexity gains across several code families (RLC, eBCH, CRC) with minimal distortion when SPC bits replace parity bits, suggesting broad applicability in energy- and latency-constrained communication systems.

Abstract

Guessing Codeword Decoding (GCD) is a recently proposed soft-input forward error correction decoder for arbitrary binary linear codes. Inspired by recent proposals that leverage binary linear codebook structure to reduce the number of queries made by Guessing Random Additive Noise Decoding (GRAND), for binary linear codes that include a full-message single parity-check (SPC) bit, we show that it is possible to reduce the number of queries made by GCD by a factor of up to 2 with the greatest guesswork reduction realized at lower SNRs, without impacting decoding precision. Codes without a full-message SPC can be modified to include one by changing a column of the generator matrix to obtain a decoding complexity advantage, and we demonstrate that this can often be done without losing decoding precision. To practically avail of the complexity advantage, a noise effect pattern generator capable of producing sequences for given Hamming weights, such as the landslide algorithm developed for ORBGRAND, is necessary.

Figures (11)

• Figure 1: Block diagram for ORBGRAND and GCD decoding of an abritrary systematic binary linear code concatenated with an SPC.
• Figure 2: Example ORB-GCD behavior vs. SA-GCD behavior for a noise realization on the $(n, k) = (7, 3)$ simplex code, with the GCD stopping condition indicated by the red line. The reliability of $Y_4$ is high, so SA-GCD pushes queries $z^{k}$ yielding $z_{k+1} = 1$ much later in the guessing order. In this example, queries with $z_{k+1} = 1$ are pushed later than the stopping condition, reducing total guesswork from 4 re-encoded sequences to 2. Note that if $|\mathrm{LLR}(Y_4)| = 0$, the guesswork of SA-GCD is identical to the guesswork of ORB-GCD because SA-GCD has no more information for guesswork ordering than ORB-GCD.
• Figure 3: For $(128, 104)$ SPC-aided RLC, SA-GCD and ORB-GCD yield identical BLER. RLC without an SPC bit performed slightly worse than SPC-aided RLC because unconstrained RLC do not guarantee that every message bit will be protected by a parity bit. The SPC bit in the modified RLC always protects every bit of the message.
• Figure 4: Average saved guesswork per bit for a $(128, 104)$RLC is shown for SA-GCD. Guesswork savings is always positive, as SA-GCD consistently performs less guesswork on average than ORB-GCD. At symbol SNR between 2 and 3 dB, where decoder error rate is below $10^{-1}$, SA-GCD typically saves around 100 guesses per message bit.
• Figure 5: Guesswork savings of SA-GCD is shown for $(128, 104)$RLC as a proportion of total guesswork for ORB-GCD. The highest guesswork proportion is saved at symbol SNR between 3 and 5 dB, where decoding involves significant guesswork but the decoder's error rate is still below $10^{-2}$.