Subcodes of Second-Order Reed-Muller Codes via Recursive Subproducts

TL;DR

This work introduces recursive subproduct codes built from RM$(1,m')$ bases to produce subcodes $\mathscr{C}^{\otimes[2,m]}$ that lie between RM$(1,mm')$ and RM$(2,mm')$ while preserving the minimum distance $d_{\min}=2^{mm'-2}$ but reducing the dimension. It provides a monomial-basis description, a closed-form minimum-weight codeword count, and weight-distribution recurrences derived from symplectic-coset analysis, enabling practical CER estimates. The authors adapt belief-propagation and local graph search decoding to exploit the product-code structure, using projections to RM$(1,d)\otimes$RM$(0,mm'-d-1)$ and extra generalized check nodes, achieving CER within 0.25 dB of RM$(2,mm')$ and outperforming CRC-aided Polar codes at lengths $256$, $512$, and $1024$. Simulation results demonstrate favorable rate-distance tradeoffs with substantially fewer minimum-weight codewords, offering a viable rate-adaptation option for RM codes in low-capacity scenarios. The paper also discusses limitations and open questions, including $m'\ge 4$ weight-distribution recurrences and potential improvements via precoding or CRC-based strategies.

Abstract

We use a simple construction called `recursive subproducts' (that is known to yield good codes of lengths $n^m$, $n \geq 3$) to identify a family of codes sandwiched between first-order and second-order Reed-Muller (RM) codes. These codes are subcodes of multidimensional product codes that use first-order RM codes as components. We identify the minimum weight codewords of all the codes in this family, and numerically determine the weight distribution of some of them. While these codes have the same minimum distance and a smaller rate than second-order RM codes, they have significantly fewer minimum weight codewords. Further, these codes can be decoded via modifications to known RM decoders which yield codeword error rates within 0.25 dB of second-order RM codes and better than CRC-aided Polar codes (in terms of $E_b/N_o$ for lengths $256, 512, 1024$), thereby offering rate adaptation options for RM codes in low-capacity scenarios.

Figures (3)

• Figure 1: Codeword error rates of codes with length $256$.
• Figure 2: Comparison of codes with length $512$.
• Figure 3: Comparison of codes with length $1024$.

Theorems & Definitions (16)

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