A Parametric Approach to List Decoding of Reed-Solomon Codes Using Interpolation

Abstract

In this paper we present a minimal list decoding algorithm for Reed-Solomon (RS) codes. Minimal list decoding for a code C refers to list decoding with radius L, where L is the minimum of the distances between the received word r and any codeword in C. We consider the problem of determining the value of L as well as determining all the codewords at distance L. Our approach involves a parametrization of interpolating polynomials of a minimal Gröbner basis G. We present two efficient ways to compute G. We also show that so-called re-encoding can be used to further reduce the complexity. We then demonstrate how our parametric approach can be solved by a computationally feasible rational curve fitting solution from a recent paper by Wu. Besides, we present an algorithm to compute the minimum multiplicity as well as the optimal values of the parameters associated with this multiplicity which results in overall savings in both memory and computation.

A module minimization approach to Gabidulin decoding via interpolation

“…This was used in [3] as the foundation for a polynomial time Reed-Solomon list decoding method via WelchBerlekamp type interpolation. In this paper we strengthened the link between Reed-Solomon decoding and Gabidulin decoding in providing a similar parametrization from a Welch-Berlekamp type algorithm for Gabidulin decoding.…”

Iterative list-decoding of Gabidulin codes via Gröbner based interpolation

“…In our algorithm we construct, via a simple update matrix, a minimal Gröbner basis at each step. This setup allows for straightforward conclusions on minimality and parametrization due to the Predictable Leading Monomial Property, as in [1] and [6].…”

“…Thus, after n steps, B n is a minimal Gröbner basis for the interpolation module M(r). Consequently, B n has the socalled Predictable Leading Monomial Property, see [6] and [1]. As a result of this property, the parametrization used for a(x) and c(x) in the second part of the algorithm will then yield all closest codewords.…”

The combination of sparse learning and list decoding of subspace codes for error correction in random network coding

“…Mahdavifar and Vardy [9] generate Silva and Kschischang code to list decoding. Especially what deserves to be mentioned most is that there are also list decoding methods based on rank distance [8] and Hamming distance [34,35]. Unlike [9] which designs list decoding with a purpose to solve the error-correcting problem in random network coding, [8,23,24,34,35] are not designed for solving the error-correcting problem in random network coding.…”

“…Unlike [9] which designs list decoding with a purpose to solve the error-correcting problem in random network coding, [8,23,24,34,35] are not designed for solving the error-correcting problem in random network coding. However, because [8,23,24,34,35] can correct errors beyond d min /2, they may have the potential to solve the propagated errors in random network coding. The fifth is the approach based on the secret channel.…”