Behavioral interpolation for coding and control

It is well-known that classical decoding of Reed-Solomon error-correcting block codes is equivalent to system theoretic minimal partial realization. In the first part of the paper we show how this type of decoding can also be formulated as minimal polynomial interpolation. We compare this type of interpolation with system-theoretic interpolation techniques that are used for control applications. We then present a procedure that achieves minimal polynomial interpolation by iteratively constructing a row reduced representation of an interpolating behavior. Motivated by the need for improved decoding techniques, in particular soft-decision decoding, we turn to “list decoding” in the second part of the paper. Here the aim is to construct a list of all code words that are within a pre-specified Hamming distance from the received word. A connection is made with work in the coding-theoretic literature that performs list decoding in two steps: in step one a minimal bivariate interpolating polynomial Q(ξ, η) is constructed whereas in step two Q(ξ, η) is factorized into factors that are linear in η. We point out that this new development opens up yet another connection between coding theory and system theory, namely the connection between list decoding and minimal multivariable interpolation