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

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 $S_d(n)={\mathrm{\Theta }}_d({\lambda }_d(n)\cdot n)$ 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 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 $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(C(x),C(y))\ge m-\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,y)+1$ 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.