Recursive Subproduct Codes with Reed-Muller-like Structure

TL;DR

This work introduces recursive subproduct codes $\mathscr{C}^{\otimes[r,m]}$, a flexible family of subcodes of $m$-fold product codes with a recursive Plotkin-like structure that admits a projection property for iterative decoding. It shows that Reed-Muller codes and Dual Berman codes arise as special cases, derives the dimension $\dim(\mathscr{C}^{\otimes[r,m]})=\sum_{l=0}^r \binom{m}{l}(k-1)^l$ and the minimum distance $d_{\min}(\mathscr{C}^{\otimes[ r,m]}) = d_{\min}(\mathscr{C})^{r} n^{m-r}$, and provides fast ML decoding for first-order codes plus a soft-output max-log-MAP decoder. For second-order codes, decoding uses a projection-based BP framework with a carefully constructed factor graph and a local graph search to boost performance; complexity scales as $\mathscr{O}(\max\{N,N^{\alpha}\})$ (or $\mathscr{O}(N \log N)$ when $\alpha=1$) per iteration. Simulations show the subproduct codes achieve better-than or within $0.5$ dB of RM and CRC-aided Polar codes across various block lengths and rates, highlighting a wide design space and practical decoding approaches for near-capacity performance.

Abstract

We study a family of subcodes of the $m$-dimensional product code $\mathscr{C}^{\otimes m}$ ('subproduct codes') that have a recursive Plotkin-like structure, and which include Reed-Muller (RM) codes and Dual Berman codes as special cases. We denote the codes in this family as $\mathscr{C}^{\otimes [r,m]}$, where $0 \leq r \leq m$ is the 'order' of the code. These codes allow a 'projection' operation that can be exploited in iterative decoding, viz., the sum of two carefully chosen subvectors of any codeword in $\mathscr{C}^{\otimes [r,m]}$ belongs to $\mathscr{C}^{\otimes [r-1,m-1]}$. Recursive subproduct codes provide a wide range of rates and block lengths compared to RM codes while possessing several of their structural properties, such as the Plotkin-like design, the projection property, and fast ML decoding of first-order codes. Our simulation results for first-order and second-order codes, that are based on a belief propagation decoder and a local graph search algorithm, show instances of subproduct codes that perform either better than or within 0.5 dB of comparable RM codes and CRC-aided Polar codes.

Figures (5)

• Figure 1: Factor graph for BP decoding of $\mathscr{C}^{\otimes[2,m]}$. The numbers next to the dashed arrows represent the steps in each iteration. Empty circles are ${\sf V}$, empty squares are ${\sf C}$, filled circles are ${\sf V}_{\sf h}$, and filled squares are ${\sf C}_{\sf g}$.
• Figure 2: Performance of first-order codes under ML decoding. 'DB' denotes Dual Berman codes, and $\mathcal{H}$ is the $[7,4,3]$ Hamming code.
• Figure 3: Performance of codes with block lengths close to $250$. Here, 'DB' denotes the Dual Berman code $\mathsf{DB}_{3}(2, 5)$, and 'CA-DB' is its CRC-aided version.
• Figure 4: Comparison of $\mathcal{H}^{\otimes[2,3]}$ with 5G New Radio CRC-aided Polar code. Both codes are of length $343$ and dimension $37$.
• Figure 5: Codes with rates close to $0.085$. Here $\textrm{DB}$ denotes the $[9,5,3]$ Dual Berman code $\mathsf{DB}_{3}(1, 2)$.

Theorems & Definitions (15)

• Example 1
• Remark 1
• Lemma 1
• Lemma 2
• Claim 1
• Remark 2
• Lemma 3
• Claim 2
• Corollary 1
• Lemma 4