We study single-error detection and correction for analog codes over $\mathbb{R}$. The key performance measures are the parameters $Γ_1(\mathcal{C})$ and $Γ_2(\mathcal{C})$, which quantify, respectively, the minimum separation required between large outlying errors that must be detected or located and the magnitude of tolerable perturbations. First, we prove that every real linear $[n,k]$ code $\mathcal{C}$ satisfies \[ Γ_1(\mathcal{C})\ge 2\left\lceil\frac{n}{n-k}\right\rceil. \] Moreover, when $k=n-2$, we prove that every real linear $[n,n-2]$ code $\mathcal{C}$ satisfies \[ Γ_2(\mathcal{C})\ge \frac{1}{\sin^2(π/2n)}. \] Together, these two lower bounds settle all four open problems of Roth concerning the optimality of single-error-detecting and single-error-correcting analog codes. The proof of the first bound is based on a double-induction argument, while the proof of the second combines a zonotope-based geometric characterization of $Γ_2(\mathcal{C})$ with a cyclic sine-product inequality. In addition, we construct analog codes with higher fixed redundancy and show that, for every fixed $r\ge 2$, there exists a class of linear $[n,\ge n-r]$ codes over $\mathbb{R}$ such that \[ Γ_2(\mathcal{C})\le O\left(n^{1+\frac{1}{r-1}}\right). \] This gives a new upper bound in the fixed-redundancy regime, which was not covered by previously known constructions.

翻译：我们研究实数域 $\mathbb{R}$ 上模拟码的单错误检测与纠正问题。关键性能指标为参数 $\Gamma_1(\mathcal{C})$ 与 $\Gamma_2(\mathcal{C})$，分别衡量需检测或定位的大离群错误间所需的最小分离度以及可容忍扰动的幅度。首先，我们证明任意实线性 $[n,k]$ 码 $\mathcal{C}$ 满足 \[ \Gamma_1(\mathcal{C})\ge 2\left\lceil\frac{n}{n-k}\right\rceil. \] 此外，当 $k=n-2$ 时，我们证明任意实线性 $[n,n-2]$ 码 $\mathcal{C}$ 满足 \[ \Gamma_2(\mathcal{C})\ge \frac{1}{\sin^2(\pi/2n)}. \] 这两个下界共同解决了 Roth 关于单错误检测与单错误纠正模拟码最优性的全部四个开放问题。第一个界的证明基于双重归纳论证，第二个界的证明则将基于 zonotope 的 $\Gamma_2(\mathcal{C})$ 几何刻画与循环正弦乘积不等式相结合。此外，我们构造了具有更高固定冗余的模拟码，并证明对于任意固定的 $r\ge 2$，存在一类 $\mathbb{R}$ 上的线性 $[n,\ge n-r]$ 码满足 \[ \Gamma_2(\mathcal{C})\le O\left(n^{1+\frac{1}{r-1}}\right). \] 这给出了固定冗余体制下的新上界，该结果未被先前的已知构造所涵盖。