Precise asymptotic wire complexity for encoding good codes at intermediate depths

Determine the precise asymptotic number of wires required by depth-d circuits with unbounded fan-in and arbitrary Boolean-function gates to encode asymptotically good error-correcting codes (codes with constant rate and constant relative distance) for super-constant depths d satisfying ω(1) ≤ d ≤ α(n) − 3, where α(n) denotes the inverse Ackermann function.

Background

The paper establishes an O(λ_d(n)·n) upper bound on the number of wires for depth-d encoding circuits of asymptotically good binary codes, and shows that linear-size encoding can be achieved at depth α(n). It also credits and refines lower-bound methods implying that, for linear-size circuits, depth at least α(n) − 2 is necessary.

However, while the methods yield superlinear lower bounds up to depth α(n) − 3, they do not determine the exact asymptotics for the required number of wires in the super-constant depth regime. Thus, the precise wire complexity for depths d growing with n but bounded by α(n) − 3 remains unresolved.