List-decoding reed-muller codes over small fields

  • Authors:

  • Parikshit Gopalan;Adam R. Klivans;David Zuckerman

  • Affiliations:

  • Univ. of Washington, Seattle, WA, USA;UT-Austin, Austin, TX, USA;UT-Austin, Austin, TX, USA

  • Venue:
  • STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
  • Year:
  • 2008

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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 --r - ε) from R for any ε r,ε-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 q, 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.