List-Decoding Algorithms for Lifted Codes


Lifted Reed–Solomon codes are a natural affine-invariant family of error-correcting codes, that generalize Reed–Muller codes. They were known to have economical native-testing and native-decoding algorithms (comparable with the known algorithms for Reed–Muller codes), however with considerably better rate. We provide economical algorithms for list decoding and local list decoding of lifted codes. Our algorithms are based on a brand new technical lemma, which says that the codewords of lifted codes are low degree polynomials when viewed as univariate polynomials over a huge field (while they'll be terribly high degree when viewed as multivariate polynomials over a small field).

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