Threshold Selection for Iterative Decoding of $(v,w)$-regular Binary Codes

TL;DR

The paper addresses efficient decoding of sparse $(v,w)$-regular binary codes by designing state-aware, syndrome-weight dependent thresholds for a two-iteration parallel bit-flipping decoder. It develops a closed-form model for the distribution of the syndrome weight after the first iteration using a non-homogeneous Markov process, enabling joint optimization of the first- and second-iteration thresholds $\mathtt{th}^{(1)}(y)$ and $\mathtt{th}^{(2)}(y,z_0,z_1)$. By deriving explicit second-iteration flip probabilities and incorporating $(v,w)$-regularity constraints, the authors present a practical two-stage threshold selection framework that achieves significantly lower decode failure rates (DFR) than fixed thresholds or BIKE-style thresholds, while remaining scalable to cryptographic code sizes. The approach offers reproducible, off-the-shelf threshold computation that improves decoding reliability for post-quantum cryptographic primitives and related applications.

Abstract

Iterative bit flipping decoders are an efficient and effective decoder choice for decoding codes which admit a sparse parity-check matrix. Among these, sparse $(v,w)$-regular codes, which include LDPC and MDPC codes are of particular interest both for efficient data correction and the design of cryptographic primitives. In attaining the decoding the choice of the bit flipping thresholds, which can be determined either statically, or during the decoder execution by using information coming from the initial syndrome value and its updates. In this work, we analyze a two-iterations parallel hard decision bit flipping decoders and propose concrete criteria for threshold determination, backed by a closed form model. In doing so, we introduce a new tightly fitting model for the distribution of the Hamming weight of the syndrome after the first decoder iteration and substantial improvements on the DFR estimation with respect to existing approaches.

Figures (5)

• Figure 1: Numerical validation of the model of syndrome weight distribution after the first iteration. Code parameters: $n=9,602$, $\frac{k}{n}=\frac{1}{2}$, $v=45$, $t=50$, $\mathtt{th}^{(1)}=25$ fixed. Numerical results obtained with $10^8$ random samples. \ref{['mark:sinweven']} depicts the model distribution over even values of $z$, \ref{['mark:sinwodd']} over odd ones.
• Figure 2: $\mathtt{DFR}^{(2)}$ for $(v,2v)$-regular LDPC codes with $\frac{k}{n}=\frac{1}{2}$, parallel decoder. Each data point is obtained with either $10^8$ decodes or reaching $100$ decoding failures. (a) Using our thresholds, our $\mathtt{DFR}^{(2)}$ model (solid lines) vs. numerical simulations (crosses); (b) Modeled $\mathtt{DFR}^{(2)}$: our thresholds (solid lines), majority thresholds $\mathtt{th}^{(1)}=\mathtt{th}^{(2)}=\lceil \frac{v+1}{2}\rceil$ (dashed lines); (c) for $v=19$ and $t=18$, performance comparison among majority thresholds (blue crosses), our choice (red crosses), best choice obtained evaluating a posteriori $10^8$ decodes on different syndromes (black circles)
• Figure 3: Average number of discrepancies left after two iteration for QC-MDPC codes with rate $\frac{k}{n}=\frac{1}{2}$, $v=71$, $t=134$, parallel decoder employing three different threshold policies: majority voting with margin $\delta=3$ ($\mathtt{th}^{(1)}=\mathtt{th}^{(2)} =\lceil \frac{v+1}{2} \rceil + \delta$), BIKE-flip threshold selection BIKE, and thresholds computed by our model. Each data point obtained performing either $10^6$ decoding actions, or reaching $10^6$ total discrepancies. $r=12,323$ corresponds to the parameter set of BIKE for NIST security level 1.
• Figure 4: Numerical validation of the model of the decoding failure rate conditioned on the syndrome weight $y$, employing the following parameter set: $n=2,990$, $\frac{k}{n}=\frac{1}{2}$, $v=23$, $t=30$, $\mathtt{th}_{(2)}=\lceil \frac{v+1}{2} \rceil$ fixed. Numerical results obtained with $10^8$ random samples for each threshold choice. Solid and dotted lines are the model, crosses are numerical simulations.
• Figure 5: Two iterations DFR estimated values for $(v,2v)$-regular LDPC codes with rate $\frac{k}{n}=\frac{1}{2}$, parallel decoder employing thresholds derived from our model (solid lines) against fixed thresholds with majority voting $\mathtt{th}^{(1)}=\mathtt{th}^{(2)}=\lceil \frac{v+1}{2} \rceil$ (dashed lines), as in DBLP:conf/isit/AnnechiniBP24, Figure 2.