Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates

  • Authors:

  • Anna Gál;Kristoffer Arnsfelt Hansen;Michal Koucký;Pavel Pudlák;Emanuele Viola

  • Affiliations:

  • University of Texas at Austin, Austin, TX, USA;Aarhus University, Aarhus, Denmark;Institute of Mathematics, the Academy of Sciences of the Czech Republic, Prague, Czech Rep;Institute of Mathematics, the Academy of Sciences of the Czech Republic, Prague, Czech Rep;Northeastern University, Boston, MA, USA

  • Venue:
  • STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2012

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Abstract

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 ({log n/ log log 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(n λk(n)), where λ1(n)=⌈ log n⌉, λi+1(n)= λi*(n), and the * operation gives how many times one has to iterate the function λi to reach a value at most 1 from the argument n. (4) If d=log* n then w=O(n). For depth d=2, our Ω(n (log n/log log n)2) lower bound gives the largest known lower bound for computing any linear map. Using a result by Ishai, Kushilevitz, Ostrovsky, and Sahai (2008), 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.