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

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F13%3A00422134" target="_blank" >RIV/67985840:_____/13:00422134 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1109/TIT.2013.2270275" target="_blank" >http://dx.doi.org/10.1109/TIT.2013.2270275</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1109/TIT.2013.2270275" target="_blank" >10.1109/TIT.2013.2270275</a>

Result language
angličtina
Original language name
Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates
Original language description
We bound the minimum number of wires needed to compute any (asymptotically good) error-correcting code C : {0, 1}(Omega(n)) -> {0, 1}(n) with minimum distance Omega(n), using unbounded fan-in circuits of depth with arbitrary gates. Our main results are:1) if d = 2, then w = Theta(n(lg n/lg lg n)(2)); 2) if d = 3, then w = Theta(n lg lg n); 3) if d = 2k or d = 2k + 1 for some integer k >= 2, then w = Theta(n lambda(k)(n)), where lambda(1)(n) = inverted rightlg ninverted left perpendicular lambda(i+1)(n)= lambda(i)*(n), and the * operation gives how many times one has to iterate the function lambda(i) to reach a value at most 1 from the argument; and 4) if d = lg* n, then w = O(n). For depth d = 2, our Omega(n(lg n/lg lg n)(2)) lower bound gives the largest known lower bound for computing any linear map. The upper bounds imply that a (necessarily dense) generator matrix for our code can be written as the product of two sparse matrices.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/IAA100190902" target="_blank" >IAA100190902: Mathematical logic, complexity, and algorithms</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2013
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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