Fast Root Finding for Interpolation-Based Decoding of Interleaved Gabidulin Codes

Abstract

We show that the root-finding step in interpolation-based decoding of interleaved Gabidulin codes can be solved by finding a so-called minimal approximant basis of a matrix over a linearized polynomial ring. Based on existing fast algorithms for computing such bases over ordinary polynomial rings, we develop fast algorithms for computing them over linearized polynomials. As a result, root finding costs $O^{\sim}(\ell^{\omega}\mathcal{M}(n))$ operations in $F_{q^{m}}$, where ℓ is the interleaving degree, n the code length, $F_{q^{m}}$ the base field of the code, $2 \leq \omega \leq 3$ the matrix multiplication exponent, and $\mathcal{M}(n) \in O(n^{1635})$ is the complexity of multiplying two linearized polynomials of degree at most n. This is an asymptotic improvement upon the previously fastest algorithm of complexity $O(\ell^{3}n^{2})$, in some cases $O(\ell^{2}n^{2})$.