### Tight Bounds on Computing Error-Correcting Codes by Bounded-Depth Circuits With Arbitrary Gates

Anna Gal, Kristoffer Arnsfelt Hansen, Michal Koucky, Pavel Pudlak, Emanuele Viola
2013 IEEE Transactions on Information Theory
We bound the minimum number w of wires needed to compute any (asymptotically good) error-correcting code C : {0, 1} Ω(n) → {0, 1} n with minimum distance Ω(n), using unbounded fan-in circuits of depth d with arbitrary gates. Our main results are: (1) If d = 2 then w = Θ(n(lg n/ lg lg n) 2 ). (2) If d = 3 then w = Θ(n lg lg n). (3) If d = 2k or d = 2k + 1 for some integer k ≥ 2 then w = Θ(nλ k (n)), where λ 1 (n) = lg n , λ i+1 (n) = λ * i (n), and the * operation gives how many times one has to
more » ... iterate the function λ i to reach a value at most 1 from the argument n. (4) If d = lg * n then w = O(n). For depth d = 2, our Ω(n(lg n/ lg lg n) 2 ) lower bound gives the largest known lower bound for computing any linear map. Using a result by Ishai, Kushilevitz, Ostrovsky, and Sahai [17], we also obtain similar bounds for computing pairwise-independent hash functions. Our lower bounds are based on a superconcentrator-like condition that the graphs of circuits computing good codes must satisfy. This condition is provably intermediate between superconcentrators and their weakenings considered before. Theorem 2 For any constants 0 < ρ ≤ 1/32, 0 < δ < 1/8 and d ≥ 1, we have w λ d (n),ρ,δ (n) = O(n). Spielman [29] achieves depth O(lg n) for linear-size circuits with bounded fan-in gates, and this is optimal for computing good codes with bounded fan-in circuits. Our theorem holds for circuits with unbounded fan-in gates.