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

2013
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IEEE Transactions on Information Theory
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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

doi:10.1109/tit.2013.2270275
fatcat:v7a6m36brzfeja4jmvdppclwau