重复测量在解码二阶里德-穆勒码中的价值
The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes
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中文总结 AI 辅助
本文比较RPA与CHIRRUP解码二阶里德-穆勒码，发现CHIRRUP在Delsarte-Goethals集合上更有效利用重复测量，优于普通RPA。
中文摘要 AI 辅助
我们比较了RPA和CHIRRUP两种解码算法在二阶里德-穆勒码上的性能。RPA算法恢复与斜对称矩阵$M$相关的二元二次型，而CHIRRUP恢复与对称矩阵$P$相关的$\mathbb{Z}_4$值二次型。我们描述了格雷映射如何连接这些形式的求值向量，从而在$M$和$P$之间诱导出保秩对应关系。我们利用Delsarte-Goethals集合$DG(m,r)$分析了欧氏距离如何由矩阵秩差控制。我们证明，在这些集合上，CHIRRUP比RPA更有效地利用重复测量。虽然一种核感知的RPA变体通过聚合$\ker M$的陪集上的投影方向，在奇异二次型上提高了性能，但CHIRRUP的树搜索在$r < \frac{m-1}{2}$时更有效地组合$DG(m,r)$上的多个行估计，优于普通RPA。
英文摘要
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 $\mathbb{Z}_4$-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 $r < \frac{m-1}{2}$, outperforming vanilla RPA.
