Place : LIX, Salle Philippe Flajolet.
Title: Nearly linear time encodable codes beating the Gilbert-Varshamov bound.
Abstract: Error-correcting codes enable reliable transmission of information over an erroneous channel. One typically desires codes to transmit information at a high rate while still being able to correct a large fraction of errors. However, rate and relative distance (which quantifies the fraction of errors corrected) are competing quantities with a trade off. The Gilbert-Varshamov bound assures for every rate R, relative distance D and alphabet size Q, there exists an infinite family of codes with R + H_Q(D) >= 1-\epsilon. Constructing codes meeting or beating the Gilbert-Varshamov bound remained a long-standing open problem, until the advent of algebraic geometry codes by Goppa. In a seminal paper, for prime power squares Q ≥ 7^2, Tsfasman-Vladut-Zink constructed algebraic geometry codes beating the Gilbert-Varshamov bound. A rare occasion where an explicit construction yields better parameters than guaranteed by randomized arguments! For codes to find use in practice, one often requires fast encoding and decoding algorithms in addition to satisfying a good trade off between rate and minimum distance. A natural question, which remains unresolved, is if there exist linear time encodable and decodable codes meeting or beating the Gilbert-Varshamov bound. In this talk, I shall present the first nearly linear time encodable codes beating the Gilbert-Varshamov bound, along with a nearly quadratic decoding algorithm. Applications to secret sharing, explicit construction of pseudorandom objects, Probabilistically Checkable Interactive Proofs and the like will also be discussed.