Improved probabilistic decoding of interleaved Reed-Solomon codes and folded Hermitian codes

  • Authors:

  • Ferruh Özbudak;Oğuz Yayla

  • Affiliations:

  • Department of Mathematics, Middle East Technical University, Dumlupinar Bul., No. 1, 06800, Ankara, Turkey and Institute of Applied Mathematics, Middle East Technical University, Dumlupinar Bul., ...;Institute of Applied Mathematics, Middle East Technical University, Dumlupinar Bul., No. 1, 06800, Ankara, Turkey

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2014

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Abstract

Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F, positive integers n, r, t and distinct elements z"1,z"2,...,z"n@?F, we present a probabilistic algorithm which can recover polynomials p"1,p"2,...,p"r@?F[x] of degree less than k"1,k"2,...,k"r respectively for a given instance "i"="1^n satisfying p"l(z"i)=y"i","l for all l@?{1,2,...,r} and for all i@?I@?{1,2,...,n} such that |I|=t with probability at least 1-n-t|F| and with time complexity at most O(rn^4) if t=max{k"1,k"2,...,k"r,n+@?"j"="1^rk"jr+1}. Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed-Solomon codes. It is observed that interleaved Reed-Solomon codes over F with rate R can be decoded up to burst error rate rr+1(1-R) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F"q"^"2"^"q with rate R can be decoded up to error rate qq+1(1-R) probabilistically.