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41st Annual Symposium on Foundations of Computer Science
"Soft-decision" decoding of Chinese remainder codes
Redondo Beach, California
November 12-November 14
ISBN: 0-7695-0850-2
V. Guruswami, Lab. for Comput. Sci., MIT, Cambridge, MA, USA
A. Sahai, Lab. for Comput. Sci., MIT, Cambridge, MA, USA
M. Sudan, Lab. for Comput. Sci., MIT, Cambridge, MA, USA
Given n relatively prime integers p/sub 1/>...>p/sub n/ and an integer k>n, the Chinese Remainder Code, CRT/sub p1,...,pnik/, has as its message space M={0,...,/spl Pi//sub i=1//sup k/,pi-1}, and encodes a message m /spl isin/M as the vector >m/sub 1/,...,m/sub n/<, where m/sub i/=m(mod p/sub i/). The soft-decision decoding problem for the Chinese remainder code is given as input a vector of residues r/spl I.oarr/=(r/sub 1/,...,r/sub n/), a vector of weights>w/sub 1/,...,w/sub n/<, and an agreement parameter t. The goal is to find all messages m /spl isin/ M such that the weighted agreement between the encoding of m and r/spl I.oarr/(i.e., /spl Sigma//sub i/ w/sub i/ summed over all i such that r/sub i/=m(mod pi)) is at least t. Here we give a new algorithm for solving the soft-decision problem for the CRT code that works provided the agreement parameter t is sufficiently large. We derive our algorithm by digging deeper into the algebra underlying the error-correcting algorithms and unveiling an "ideal"-theoretic view of decoding. When all weights are equal to 1, we obtain the more commonly studied "list decoding" problem. List decoding algorithms for the Chinese Remainder Code were given recently by O. Goldreich et al. (1999), and improved by D. Boneh. Their algorithms work for t/spl ges//spl radic/(2knlogp/sub n//logp1) and t/spl ges//spl radic/(knlogp/sub n//logp/sub 1/), respectively. We improve upon the algorithms above by using our soft-decision decoding algorithm with a non-trivial choice of weights, solve the list decoding problem provided t/spl ges//spl radic/(k(n+/spl epsi/)), for arbitrarily small /spl epsi//spl ges/0.
Index Terms:
decoding; encoding; soft-decision decoding; Chinese remainder codes; prime integers; message space; agreement parameter; encoding; error-correcting algorithms; list decoding
V. Guruswami, A. Sahai, M. Sudan, ""Soft-decision" decoding of Chinese remainder codes," focs, pp.159, 41st Annual Symposium on Foundations of Computer Science, 2000
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