Sparse Polynomial Interpolation Codes and their Decoding Beyond Half the Minimum Distance * Erich L. Kaltofen Clément Pernet Dept. of Mathematics, NCSU U. J. Fourier, LIP-AriC, CNRS, Inria, UCBL, Raleigh, NC 27695, USA ÉNS de Lyon
[email protected] 46 Allée d’Italie, 69364 Lyon Cedex 7, France www4.ncsu.edu/~kaltofen
[email protected] http://membres-liglab.imag.fr/pernet/ ABSTRACT General Terms: Algorithms, Reliability We present algorithms performing sparse univariate pol- Keywords: sparse polynomial interpolation, Blahut's ynomial interpolation with errors in the evaluations of algorithm, Prony's algorithm, exact polynomial fitting the polynomial. Based on the initial work by Comer, with errors. 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 1. INTRODUCTION divided by twice the sparsity. At ISSAC 2012, we have Evaluation-interpolation schemes are a key ingredient given a decoding algorithm for as much as half the min- in many of today's computations. Model fitting for em- imal distance and a list decoding algorithm up to the pirical data sets is a well-known one, where additional minimal distance. information on the model helps improving the fit. In Our new polynomial-time list decoding algorithm uses particular, models of natural phenomena often happen sub-sequences of the received evaluations indexed by to be sparse, which has motivated a wide range of re- an arithmetic progression, allowing the decoding for a search including compressive sensing [4], and sparse in- larger radius, that is, more errors in the evaluations terpolation of polynomials [23,1, 15, 13,9, 11].