Optimal Exponent of the Single-Error Correction Threshold with Fixed Redundancy for Analog Error-Correcting Codes

Abstract

Analog error-correcting codes (Analog ECCs), introduced by Roth [1], address errors in vector-matrix multiplication arising from analog noise and sparse outliers in in-memory computing. A fundamental open problem concerns the lower bound on the single-error correction threshold $\Gamma_2(\mathcal C)$ for real $[n,k]$ linear codes with fixed redundancy $r=n-k\geq 2$. Li et al. [2] recently established that for redundancy $r=2$, every real $[n,n-2]$ linear code $\mathcal{C}$ satisfies $\Gamma_2(\mathcal C)\geq \csc^2(\frac{\pi}{2n})$, resolving an open problem in [1], and showed that, for every fixed $r \geq 2$, there exists a class of $[n,k]$ linear code $\mathcal{C}$ over $\mathbb{R}$ such that $\Gamma_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})$. This paper proves the matching converse in [2]. For every $[n,k]$ linear code $\mathcal{C}\subseteq \mathbb R^n$ with fixed redundancy $2\leq r<n$, we show that \[ \Gamma_2(\mathcal C)\ge \frac{a_r}{\sqrt{r}\,\beta_{r-1}\,2^{\frac{1}{r-1}}}\cdot n^{1+\frac{1}{r-1}}, \] where $a_r =\frac{\Gamma(\frac{r}{2})}{\sqrt{\pi}\,\Gamma(\frac{r+1}{2})}$ and $\beta_d = \left(\frac{d\pi^{d-1}|\mathbb S^d|}{|\mathbb S^{d-1}|}\right)^{\frac{1}{d}}$ for positive integer $d$. Here $\mathbb S^d$ denotes the unit sphere in $\mathbb R^{d+1}$, $|\mathbb S^d|$ its surface area, and $\Gamma(\cdot)$ the Gamma function. In particular, we further show that $\Gamma_2(\mathcal C)\geq \frac{1}{4\pi\sqrt{3}r}\cdot n^{1+\frac{1}{r-1}}$. Together with the upper bound in [2], this confirms that the exponent $n^{1+\frac{1}{r-1}}$ is optimal, completing the asymptotic characterization of the single-error correction threshold for Analog ECCs.