Chinese remaindering with errors

The Chinese remainder theorem states that a positive integer m is uniquely specified by its remainder module k relatively prime integers p ₁, ···, p_k, provided m<Π_i=1^kp_i. Thus the residues of m module relatively prime integers p₁<p₂<···<p_n form a redundant representation of m if m<Π_i=1 ^kp_i and k<n. This gives a number-theoretic construction of an “error-correcting code” that has been considered often in the past. In this code a “message” (integer) m<Π_i=1^kp_i is encoded by the list of its residues module p₁, ···, p_n. By the Chinese remainder theorem, if a codeword is corrupted in e<(n-k)/2 coordinates, then there exists a unique integer m whose corresponding codesword differs from the corrupted word in at most e places. Furthermore, Mandelbaum (1976, 1978) shows how m can be recovered efficiently given the corrupted word provided that the p_is are very close to one another. To deal with arbitrary p _is, we present a variant of his algorithm that runs in almost linear time and recovers from e<(log p₁)/(log p₁+log p_n)·(n-k) errors. Our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than (n-k)/2. Specifically, given n residues r ₁, ···, r_n and an agreement parameter t, we find a list of all integers m<Π_i=1^kp_i such that (m mod p_i)=r_i for at least t values of i∈{1, ···, n}, provided t=Ω(√(kn(log p_n /log p₁))). For n≫k (and p_n⩽p₁ ^O(1)), the fraction of error corrected by the algorithm is almost twice that corrected by the previous work. More significantly, the algorithm recovers the message even when the amount of agreement between the “received word” and the “codeword” is much smaller than the number of errors