Limits to list decoding Reed-Solomon codes

• Authors: Venkatesan Guruswami;Atri Rudra
• Venue: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
• Year: 2005

In this paper, we prove the following two results that expose some combinatorial limitations to list decoding Reed-Solomon codes.Given n distinct elements α1,...,αn from a field F, and n subsets S1,...,Sn of F each of size at most l, the list decoding algorithm of Guruswami and Sudan [7] can in polynomial time output all polynomials p of degree at most k which satisfy p(αi) ∈ Si for every i, as long as l i,∈i) ∈ F2 (the βi's need not be distinct), find and output all degree k polynomials p such that p(βi) = γi for at least $t$ values of i, provided t √k n'. By our result, an improvement to the Reed-Solomon list decoder of [7] that works with slightly smaller agreement, say t √kn' - k/2, can only be obtained by exploiting some property of the βi's (for example, their (near) distinctness).For Reed-Solomon codes of block length $n$ and dimension k where k = nδ for small enough δ, we exhibit an explicit received word r with a super-polynomial number of Reed-Solomon codewords that agree with it on $(2 - ε) k locations, for any desired ε 0 (we note agreement of k is trivial to achieve). Such a bound was known earlier only for a non-explicit center. We remark that finding explicit bad list decoding configurations is of significant interest --- for example the best known rate vs. distance trade-off is based on a bad list decoding configuration for algebraic-geometric codes [14] which is unfortunately not explicitly known.