NDLI: Sparse polynomial interpolation codes and their decoding beyond half the minimum distance

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Sparse polynomial interpolation codes and their decoding beyond half the minimum distance

Content Provider	ACM Digital Library
Author	Kaltofen, Erich L. Pernet, Clément
Abstract	We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Based on the initial work by Comer, Kaltofen and Pernet [Proc. ISSAC 2012], we define the sparse polynomial interpolation codes and state that their minimal distance is precisely the code-word length divided by twice the sparsity. At ISSAC 2012, we have given a decoding algorithm for as much as half the minimal distance and a list decoding algorithm up to the minimal distance. Our new polynomial-time list decoding algorithm uses sub-sequences of the received evaluations indexed by an arithmetic progression, allowing the decoding for a larger radius, that is, more errors in the evaluations while returning a list of candidate sparse polynomials. We quantify this improvement for all typically small values of number of terms and number of errors, and provide a worst case asymptotic analysis of this improvement. For instance, for sparsity T = 5 with ≤ 10 errors we can list decode in polynomial-time from 74 values of the polynomial with unknown terms, whereas our earlier algorithm required 2T(E + 1) = 110 evaluations. We then propose two variations of these codes in characteristic zero, where appropriate choices of values for the variable yield a much larger minimal distance: the code-word length minus twice the sparsity.
Starting Page	272
Ending Page	279
Page Count	8
File Format	PDF
ISBN	9781450325011
DOI	10.1145/2608628.2608660
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-07-23
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Prony's algorithm Exact polynomial fitting with errors Sparse polynomial interpolation Blahut's algorithm
Content Type	Text
Resource Type	Article

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