TY - JOUR
T1 - Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels
AU - Shaltiel, Ronen
AU - Silbak, Jad
N1 - Funding Information:
We are grateful to Swastik Kopparty for pointing us to the Algebraic Geometric codes of Garcia and Stichtenoth, and in particular for pointing us to their description in Shpilka (2009). We thank Noga Ron-Zewi for help with list-recoverable codes. We also thank the anonymous referees for helpful feedback. Ronen Shaltiel was supported by ERC starting grant 279559, ISF grant 864/11, ISF grant 1628/17 and BSF grant 2010120. Jad Silbak was supported by ERC starting grant 279559 and ISF grant 1628/17. We are grateful to anonymous referees for helpful comments and corrections.
Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021/6
Y1 - 2021/6
N2 - A stochastic code is a pair of encoding and decodingprocedures (Enc, Dec) where Enc : { 0 , 1 } k× { 0 , 1 } d→ { 0 , 1 } n. Thecode is (p, L)-list decodable against a class C of “channel functions”C: { 0 , 1 } n→ { 0 , 1 } n if for every message m∈ { 0 , 1 } k and every channelC∈ C that induces at most pn errors, applying Dec on the “receivedword” C(Enc(m,S)) produces a list of at most L messages that containm with high probability over the choice of uniform S← { 0 , 1 } d. Notethat both the channel C and the decoding algorithm Dec do not receivethe random variable S, when attempting to decode. The rate of a codeis R= k/ n, and a code is explicit if Enc, Dec run in time poly(n). Guruswami and Smith (Journal of the ACM, 2016) showed that forevery constants 00 and c> 1 there exist a constantL and a Monte Carlo explicit constructions of stochastic codes withrate R≥ 1 - H(p) - ϵ that are (p, L)-list decodable for size nc channels.Here, Monte Carlo means that the encoding and decoding need to sharea public uniformly chosen poly (nc) bit string Y, and the constructedstochastic code is (p, L)-list decodable with high probability over thechoice of Y. Guruswami and Smith pose an open problem to give fully explicit (thatis not Monte Carlo) explicit codes with the same parameters, underhardness assumptions. In this paper, we resolve this open problem,using a minimal assumption: the existence of poly-time computablepseudorandom generators for small circuits, which follows from standardcomplexity assumptions by Impagliazzo and Wigderson (STOC97). Guruswami and Smith also asked to give a fully explicit unconditionalconstructions with the same parameters against O(log n) -space onlinechannels. (These are channels that have space O(log n) and are allowedto read the input codeword in one pass.) We also resolve this openproblem. Finally, we consider a tighter notion of explicitness, in which the runningtime of encoding and list-decoding algorithms does not increase, whenincreasing the complexity of the channel. We give explicit constructions(with rate approaching 1 - H(p) for every p≤ p for some p> 0) forchannels that are circuits of size 2nΩ(1/d) and depth d. Here, the runningtime of encoding and decoding is a polynomial that does not depend onthe depth of the circuit. Our approach builds on the machinery developed by Guruswami andSmith, replacing some probabilistic arguments with explicit constructions.We also present a simplified and general approach that makesthe reductions in the proof more efficient, so that we can handle weakclasses of channels.
AB - A stochastic code is a pair of encoding and decodingprocedures (Enc, Dec) where Enc : { 0 , 1 } k× { 0 , 1 } d→ { 0 , 1 } n. Thecode is (p, L)-list decodable against a class C of “channel functions”C: { 0 , 1 } n→ { 0 , 1 } n if for every message m∈ { 0 , 1 } k and every channelC∈ C that induces at most pn errors, applying Dec on the “receivedword” C(Enc(m,S)) produces a list of at most L messages that containm with high probability over the choice of uniform S← { 0 , 1 } d. Notethat both the channel C and the decoding algorithm Dec do not receivethe random variable S, when attempting to decode. The rate of a codeis R= k/ n, and a code is explicit if Enc, Dec run in time poly(n). Guruswami and Smith (Journal of the ACM, 2016) showed that forevery constants 00 and c> 1 there exist a constantL and a Monte Carlo explicit constructions of stochastic codes withrate R≥ 1 - H(p) - ϵ that are (p, L)-list decodable for size nc channels.Here, Monte Carlo means that the encoding and decoding need to sharea public uniformly chosen poly (nc) bit string Y, and the constructedstochastic code is (p, L)-list decodable with high probability over thechoice of Y. Guruswami and Smith pose an open problem to give fully explicit (thatis not Monte Carlo) explicit codes with the same parameters, underhardness assumptions. In this paper, we resolve this open problem,using a minimal assumption: the existence of poly-time computablepseudorandom generators for small circuits, which follows from standardcomplexity assumptions by Impagliazzo and Wigderson (STOC97). Guruswami and Smith also asked to give a fully explicit unconditionalconstructions with the same parameters against O(log n) -space onlinechannels. (These are channels that have space O(log n) and are allowedto read the input codeword in one pass.) We also resolve this openproblem. Finally, we consider a tighter notion of explicitness, in which the runningtime of encoding and list-decoding algorithms does not increase, whenincreasing the complexity of the channel. We give explicit constructions(with rate approaching 1 - H(p) for every p≤ p for some p> 0) forchannels that are circuits of size 2nΩ(1/d) and depth d. Here, the runningtime of encoding and decoding is a polynomial that does not depend onthe depth of the circuit. Our approach builds on the machinery developed by Guruswami andSmith, replacing some probabilistic arguments with explicit constructions.We also present a simplified and general approach that makesthe reductions in the proof more efficient, so that we can handle weakclasses of channels.
KW - 68Q01
KW - computationally bounded channels
KW - error correcting codes
KW - pseudorandomness
UR - http://www.scopus.com/inward/record.url?scp=85099585325&partnerID=8YFLogxK
U2 - 10.1007/s00037-020-00203-w
DO - 10.1007/s00037-020-00203-w
M3 - Article
AN - SCOPUS:85099585325
SN - 1016-3328
VL - 30
JO - Computational Complexity
JF - Computational Complexity
IS - 1
M1 - 3
ER -