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

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  • Anna Gál, United States
  • Kristoffer Arnsfelt Hansen
  • Michal Koucký, Czech Republic
  • Pavel Pudlák, Czech Republic
  • Emanuele Viola, United States
We bound the minimum number w of wires needed to compute any (asymptotically good) error-correcting code
C:01(n)01n with minimum distance (n),
using unbounded fan-in circuits of depth d with arbitrary gates. Our main results are:

(1) If d=2 then w=(n(lognloglogn)2) .

(2) If d=3 then w=(nlglgn).

(3) If d=2k or d=2k+1 for some integer k2 then w=(nk(n)), where 1(n)=logn , i+1(n)=i(n), and the operation gives how many times one has to iterate the function i to reach a value at most 1 from the argument n.

(4) If d=logn then w=O(n).

Each bound is obtained for the first time in this paper.
For depth d=2,
our (n(lognloglogn)2) lower bound gives the
largest known lower bound for computing any linear map,
improving on the (nlg32n) bound of Pudlak and Rodl (Discrete Mathematics '94).

We find the upper bounds surprising.
They imply that a (necessarily dense) generator matrix
for the code can be written as the product of two sparse matrices.
The upper bounds are non-explicit: we show the existence of
circuits (consisting of only XOR gates) computing good codes
within the stated bounds.

Using a result by Ishai, Kushilevitz, Ostrovsky, and Sahai (STOC '08),
we also obtain similar bounds for computing pairwise-independent hash
functions.

Furthermore, we identify a new class of superconcentrator-like graphs with connectivity properties distinct from previously-studied ones.
Original languageEnglish
JournalElectronic Colloquium on Computational Complexity
Volume18
Issue150
Number of pages33
ISSN1433-8092
Publication statusPublished - 2011

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