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List-decoding reed-muller codes over small fields

Published: 17 May 2008 Publication History

Abstract

We present the first local list-decoding algorithm for the rth order Reed-Muller code RM(2,m) over F for r ≥ 2. Given an oracle for a received word R: Fm -< F, our randomized local list-decoding algorithm produces a list containing all degree r polynomials within relative distance (2-r - ε) from R for any ε < 0 in time poly(mr-r). The list size could be exponential in m at radius 2-r, so our bound is optimal in the local setting. Since RM(2,m) has relative distance 2-r, our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed running-time polynomial in the block-length, we show that list-decoding is possible up to even larger radii, beyond the minimum distance. We give a deterministic list-decoder that works at error rate below J(21-r), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2,m) codes are list-decodable up to radius η for any constant η < 1/2 in time polynomial in the block-length. Over small fields Fq, we present list-decoding algorithms in both the global and local settings that work up to the list-decoding radius. We conjecture that the list-decoding radius approaches the minimum distance (like over F), and prove this holds true when the degree is divisible by q-1.

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    cover image ACM Conferences
    STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
    May 2008
    712 pages
    ISBN:9781605580470
    DOI:10.1145/1374376
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    Published: 17 May 2008

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    Author Tags

    1. fitting polynomials
    2. list-decoding
    3. reed-muller codes
    4. self-correctors

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    May 17 - 20, 2008
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