Multiplicity assignments for algebraic soft-decoding of Reed-Solomon codes using the method of types

Abstract

The probability of error in the Koetter-Vardy algebraic soft-decoding algorithm for Reed-Solomon codes is determined by the multiplicity assignment scheme used. A multiplicity assignment scheme converts the reliability matrix Π, consisting of the probabilities observed at the channel output, into a multiplicity matrix M that specifies the algebraic interpolation conditions. Using the method of types, Sanov's theorem in particular, we obtain tight exponential bounds on the probability of decoding error for a given multiplicity matrix. These bounds turn out to be essentially the same as the Chernoff bound. We establish several interesting properties of the multiplicity matrix M† which minimizes the exponent of the probability of error. Based on these observations, we develop a low-complexity multiplicity assignment scheme which uses nested bisection to solve for M†. This scheme provides the same probability of error as a known scheme based upon the Chernoff bound, but with much lower complexity. We also derive a simple condition on the reliability matrix Π which guarantees an exponentially small probability of error. This condition is akin to an error-correction radius, and can be used to study the performance of algebraic soft-decoding.