The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes
Abstract
We compare the performance of the RPA and CHIRRUP decoding algorithms for second-order Reed--Muller codes. The RPA algorithm recovers a binary quadratic form associated with a skew-symmetric matrix M, whereas CHIRRUP recovers a ℤ4-valued quadratic form associated with a symmetric matrix P. We describe how the Gray map connects the evaluation vectors of these forms, inducing a rank-preserving correspondence between M and P. We analyze how Euclidean distance between evaluation vectors is governed by matrix rank differences using Delsarte--Goethals sets DG(m,r). We demonstrate that CHIRRUP uses repeated measurements more effectively than RPA on these ensembles. While a kernel-aware variant of RPA improves performance on singular quadratic forms by aggregating projection directions across cosets of ker M, CHIRRUP's tree search combines multiple row estimates more efficiently on DG(m,r) when r < (m−1)/2, outperforming vanilla RPA.
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BibTeX
@misc{gao2026value,
title = {{The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes}},
author = {Ziang Gao and Robert Calderbank},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24647},
eprint = {2609.24647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24647},
}
RIS
TY - UNPB
TI - The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes
AU - Ziang Gao
AU - Robert Calderbank
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24647
UR - https://arxiv.org/abs/2609.24647
AB - We compare the performance of the RPA and CHIRRUP decoding algorithms for second-order Reed--Muller codes. The RPA algorithm recovers a binary quadratic form associated with a skew-symmetric matrix M, whereas CHIRRUP recovers a ℤ_4-valued quadratic form associated with a symmetric matrix P. We describe how the Gray map connects the evaluation vectors of these forms, inducing a rank-preserving correspondence between M and P. We analyze how Euclidean distance between evaluation vectors is governed by matrix rank differences using Delsarte--Goethals sets DG(m,r). We demonstrate that CHIRRUP uses repeated measurements more effectively than RPA on these ensembles. While a kernel-aware variant of RPA improves performance on singular quadratic forms by aggregating projection directions across cosets of ker M, CHIRRUP's tree search combines multiple row estimates more efficiently on DG(m,r) when r < (m−1)/2, outperforming vanilla RPA.
ER -