On the Minimum Depth of Circuits with Linear Number of Wires Encoding Good Codes

TL;DR

The paper tackles the problem of encoding asymptotically good error-correcting codes with depth-$d$ circuits using a linear number of wires, formalized as $S_d(n)$. It proves an upper bound $S_d(n)=O(\lambda_d(n)\cdot n)$ for all $d\ge3$, and, at depth $d=\alpha(n)$, achieves linear size $O(n)$, improving prior analyses that had larger hidden constants; the depth lower bound from previous work yields $\alpha(n)-2$, making the bound near-tight. A key technical contribution is a refined, absolute-constant construction that combines partial good codes via range detectors, condensers, amplifiers, and a composition lemma, avoiding uncontrolled growth in constants and enabling depth-$\alpha(n)$ linear-size encodings. The paper also develops and exploits superconcentrator-induced codes, establishing a two-way correspondence: superconcentrators can be used to build good codes over large fields, and any circuit encoding such a code must themselves realize a superconcentrator, linking circuit lower bounds to graph-theoretic constructions. Together, these results advance the understanding of the depth–size trade-off for encoding good codes and illuminate connections to superconcentrators and related coding-theoretic structures.

Abstract

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 $4n$. Gál, Hansen, Koucký, Pudlák, and Viola [IEEE Trans. Inform. Theory 59(10), 2013] proved that $S_d(n) = Θ_d(λ_d(n)\cdot n)$ for any fixed $d \ge 3$. By improving their construction and analysis, we prove $S_d(n)= O(λ_d(n)\cdot n)$. Letting $d = α(n)$, a version of the inverse Ackermann function, we obtain circuits of linear size. This depth $α(n)$ is the minimum possible to within an additive constant 2; we credit the nearly-matching depth lower bound to Gál 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 Pudlák [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.

Figures (3)

• Figure 1: Gate in a probabilistic circuit
• Figure 2: Partial good codes composition
• Figure 3: Inductive proof of Theorem \ref{['thm:ub_main']}

Theorems & Definitions (59)

• theorem 1
• theorem 2
• theorem 3
• definition 1
• definition 2: Inverse Ackermann Function
• definition 3
• definition 4
• definition 5
• theorem 4
• lemma 1