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Limits to List Decoding Reed–Solomon Codes

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2 Author(s)
Guruswami, V. ; Dept. of Comput. Sci. & Eng., Univ. of Washington, Seattle, WA ; Rudra, A.

In this paper, we prove the following two results that expose some combinatorial limitations to list decoding Reed-Solomon codes. 1) Given n distinct elements alpha1,...,alphan 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 can in polynomial time output all polynomials p of degree at most k that satisfy p(alphai)isinSi for every i, as long as l<lceiln/krceil. We show that the performance of this algorithm is the best possible in a strong sense; specifically, when l=lceiln/krceil, the list of output polynomials can be superpolynomially large in n. 2) For Reed-Solomon codes of block length n and dimension k+1 where k=ndelta for small enough delta, we exhibit an explicit received word with a superpolynomial number of Reed-Solomon codewords that agree with it on (2-epsi)k locations, for any desired epsi>0 (agreement of k is trivial to achieve). Such a bound was known earlier only for a nonexplicit center. Finding explicit bad list decoding configurations is of significant interest-for example, the best known rate versus distance tradeoff, due to Xing, is based on a bad list decoding configuration for algebraic-geometric codes, which is unfortunately not explicitly known

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Information Theory, IEEE Transactions on  (Volume:52 ,  Issue: 8 )