Lecture 6: Local list decoding of polynomial codes
نویسندگان
چکیده
منابع مشابه
Some remarks on multiplicity codes
Multiplicity codes are algebraic error-correcting codes generalizing classical polynomial evaluation codes, and are based on evaluating polynomials and their derivatives. This small augmentation confers upon them better local decoding, list-decoding and local list-decoding algorithms than their classical counterparts. We survey what is known about these codes, present some variations and improv...
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We briefly survey some recent progress on list decoding algorithms for binary codes. The results discussed include: – Algorithms to list decode binary Reed-Muller codes of any order up to the minimum distance, generalizing the classical GoldreichLevin algorithm for RM codes of order 1 (Hadamard codes). These algorithms are “local” and run in time polynomial in the message length. – Construction...
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We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching 1 while simultaneously allowing for sublinear-time error-correction. In this paper, we show that multiplicity codes also admit powerful list-decoding and local list-decoding algorithms correcting a large fraction of errors. Sta...
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This paper is concerned with a new family of error-correcting codes based on algebraic curves over finite fields, and list decoding algorithms for them. The basic goal in the subject of list decoding is to construct error-correcting codes C over some alphabet Σ which have good rate R, and at the same every Hamming ball of (relative) radius p has few codewords of C, and moreover these codewords ...
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We consider the question of decoding Reed-Muller codes over Fn 2 beyond their listdecoding radius. Since, by definition, in this regime one cannot demand an efficient exact list-decoder, we seek an approximate decoder: Given a word F and radii r′ > r > 0, the goal is to output a codeword within radius r′ of F, if there exists a codeword within distance r. As opposed to the list decoding problem...
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