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

TL;DR

This paper introduces a new family of codes between first- and second-order Reed-Muller codes using recursive subproducts, with improved decoding performance and fewer minimum weight codewords, suitable for low-capacity scenarios.

Contribution

The paper identifies a novel family of subcodes of second-order Reed-Muller codes constructed via recursive subproducts, with detailed weight distribution and enhanced decoding performance.

Findings

Codes have same minimum distance but fewer minimum weight codewords.

Decoding modifications achieve error rates close to second-order RM codes.

Performance surpasses CRC-aided Polar codes at certain lengths.

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.