Let $S_d(n)$ denote the minimum number of wires of a depth-$d$ (unbounded fan-in) circuit encoding an error-correcting code $C:\{0, 1\}^n \to \{0, 1\}^{32n}$ with distance at least $4n$. G\'{a}l, Hansen, Kouck\'{y}, Pudl\'{a}k, and Viola [IEEE Trans. Inform. Theory 59(10), 2013] proved that $S_d(n) = \Theta_d(\lambda_d(n)\cdot n)$ for any fixed $d \ge 3$. By improving their construction and analysis, we prove $S_d(n)= O(\lambda_d(n)\cdot n)$. Letting $d = \alpha(n)$, a version of the inverse Ackermann function, we obtain circuits of linear size. This depth $\alpha(n)$ is the minimum possible to within an additive constant 2; we credit the nearly-matching depth lower bound to G\'{a}l et al., since it directly follows their method (although not explicitly claimed or fully verified in that work), and is obtained by making some constants explicit in a graph-theoretic lemma of Pudl\'{a}k [Combinatorica, 14(2), 1994], extending it to super-constant depths. We also study a subclass of MDS codes $C: \mathbb{F}^n \to \mathbb{F}^m$ characterized by the Hamming-distance relation $\mathrm{dist}(C(x), C(y)) \ge m - \mathrm{dist}(x, y) + 1$ for any distinct $x, y \in \mathbb{F}^n$. (For linear codes this is equivalent to the generator matrix being totally invertible.) We call these superconcentrator-induced codes, and we show their tight connection with superconcentrators. Specifically, we observe that any linear or nonlinear circuit encoding a superconcentrator-induced code must be a superconcentrator graph, and any superconcentrator graph can be converted to a linear circuit, over a sufficiently large field (exponential in the size of the graph), encoding a superconcentrator-induced code.

翻译：设 $S_d(n)$ 表示深度为 $d$（无扇入限制）的电路编码纠错码 $C:\{0, 1\}^n \to \{0, 1\}^{32n}$（距离至少为 $4n$）所需的最小线数。Gál、Hansen、Koucký、Pudlák 和 Viola [IEEE Trans. Inform. Theory 59(10), 2013] 证明，对于任意固定的 $d \ge 3$，有 $S_d(n) = \Theta_d(\lambda_d(n)\cdot n)$。通过改进他们的构造和分析，我们证明 $S_d(n)= O(\lambda_d(n)\cdot n)$。令 $d = \alpha(n)$（反阿克曼函数的一个版本），我们得到了线性规模的电路。这个深度 $\alpha(n)$ 是在加法常数 2 范围内的最小可能深度；我们将近乎匹配的深度下界归功于 Gál 等人，因为它直接遵循他们的方法（尽管在该工作中未明确声称或完全验证），并且通过将 Pudlák [Combinatorica, 14(2), 1994] 的一个图论引理中的一些常数显式化，并将其扩展到超常数深度而得到。我们还研究了一类 MDS 码 $C: \mathbb{F}^n \to \mathbb{F}^m$，其特点是对任意不同的 $x, y \in \mathbb{F}^n$ 满足汉明距离关系 $\mathrm{dist}(C(x), C(y)) \ge m - \mathrm{dist}(x, y) + 1$。（对于线性码，这等价于生成矩阵是完全可逆的。）我们称这些码为超集中诱导码，并证明了它们与超集中器的紧密联系。具体来说，我们观察到任何编码超集中诱导码的线性或非线性电路都必须是超集中器图，而任何超集中器图都可以转换为一个在足够大的域（指数级于图的大小）上编码超集中诱导码的线性电路。