A Random Coding Approach to Performance Analysis of the Ordered Statistic Decoding with Local Constraints

TL;DR

LC-OSD extends OSD by exploiting local constraints to form a MRB-based trellis, reducing the search burden while retaining near-ML performance. A random-coding analysis bounds the LC-OSD performance gap to ML and guides tuning of the constraint degree $\delta$ and list size $\ell_{\max}$, complemented by practical stopping criteria (DAI/SAI) to cut computations with negligible performance loss. The paper introduces two LGAs, SLVA and tFPT, with well-characterized time/space complexities, and demonstrates through simulations (including RS and CA-polar codes) that LC-OSD can approach RC-U bounds and offer high throughput in high-SNR regimes. Overall, the framework provides a universal, parameterizable decoding approach with predictable performance bounds and scalable complexity suitable for short-block codes and finite-length regimes.

Abstract

This paper is concerned with the ordered statistic decoding with local constraints (LC-OSD) of binary linear block codes, which is a near maximum-likelihood decoding algorithm. Compared with the conventional OSD, the LC-OSD significantly reduces both the maximum and the average number of searches. The former is achieved by performing the serial list Viterbi algorithm (SLVA) or a two-way flipping pattern tree (FPT) algorithm with local constraints on the test error patterns, while the latter is achieved by incorporating tailored early termination criteria. The main objective of this paper is to explore the relationship between the performance of the LC-OSD and decoding parameters, such as the constraint degree and the maximum list size. To this end, we approximate the local parity-check matrix as a totally random matrix and then estimate the performance of the LC-OSD by analyzing with a saddlepoint approach the performance of random codes over the channels associated with the most reliable bits (MRBs). The random coding approach enables us to derive an upper bound on the performance and predict the average rank of the transmitted codeword in the list delivered by the LC-OSD. This allows us to balance the constraint degree and the maximum list size for the average (or maximum) time complexity reduction. Simulation results show that the approximation by random coding approach is numerically effective and powerful. Simulation results also show that the RS codes decoded by the LC-OSD can approach the random coding union (RCU) bounds, verifying the efficiency and universality of the LC-OSD.

Figures (16)

• Figure 1: Numerical illustration of the relationships among soft weights $\Gamma(\cdot)$. The data are obtained by simulating a $\mathscr{C}[128, 64]$ code at $E_{\textrm{b}}/N_0 = 2.5~\textrm{dB}$, where the LC-OSD parameters are $(\delta, \ell_{\textrm{max}}) = (8, 2^{7})$.
• Figure 2: Sketches of $(n-k-\delta)$-th $n$-quantile for $|r|$, where $(n, k, \delta) = (128, 64, 8)$ and $E_{\textrm{b}}/N_0=2.0~\textrm{dB}$.
• Figure 3: Simulation results of the LDPC code $\mathscr{C}_{\textrm{LDPC}}[128,64]$ constructed in Baldi2016OSD.
• Figure 4: Simulation results of the RS codes $\mathscr{C}_{\textrm{RS}}[31, k]_{2^5}$, which are mapped into $\mathbb{F}_2^{155}$ for LC-OSD$(8, 2^{14})$.
• Figure 5: The CCDF of $D(\bm{r})$, where the randomness comes from the LLR vector $\bm{r}$. Here $(n, k, \delta) = (128, 64, 8)$.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Proposition 1
• Definition 3
• Proposition 2
• Definition 4
• Proposition 3
• Definition 5
• Proposition 4
• Definition 6