“Soft-decision” decoding of Chinese remainder codes

Given n relatively prime integers p₁<...<p_n and an integer k<n, the Chinese Remainder Code, CRT_p1,...,pnik, has as its message space M={0,...,Π_i=1^k,pi-1}, and encodes a message m ∈M as the vector ⟨m₁,...,m_n⟩, where m_i=m(mod p_i). The soft-decision decoding problem for the Chinese remainder code is given as input a vector of residues r&oarr;=(r₁,...,r_n), a vector of weights ⟨w ₁,...,w_n⟩, and an agreement parameter t. The goal is to find all messages m ∈ M such that the weighted agreement between the encoding of m and r&oarr;(i.e., Σ_i w_i summed over all i such that r_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⩾√(2knlogp _n/logp1) and t⩾√(knlogp_n/logp₁), 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⩾√(k(n+ε)), for arbitrarily small ε⩾0