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

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
^~
(ℓ
^ω
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 ≤ ω ≤ 3 the matrix multiplication exponent, and M(n) ∈ O(n
^1.635
) 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(ℓ
³
n
²
), in some cases O(ℓ
²
n
²
).