List and unique error-erasure decoding of interleaved Gabidulin codes with interpolation techniques

Abstract

A new interpolation-based decoding principle for interleaved Gabidulin codes is presented. The approach consists of two steps: First, a multi-variate linearized polynomial is constructed which interpolates the coefficients of the received word and second, the roots of this polynomial have to be found. Due to the specific structure of the interpolation polynomial, both steps (interpolation and root-finding) can be accomplished by solving a linear system of equations. This decoding principle can be applied as a list decoding algorithm (where the list size is not necessarily bounded polynomially) as well as an efficient probabilistic unique decoding algorithm. For the unique decoder, we show a connection to known unique decoding approaches and give an upper bound on the failure probability. Finally, we generalize our approach to incorporate not only errors, but also row and column erasures.

Appendix

Lemma 12

(Row Space of Composition)

Let ${a}(x)$ and ${b}(x)$ denote two linearized polynomials in $\mathbb {L}_{q^m}[x]$ with $\deg _q {a}(x),$ $\deg _q {b}(x) < m$. Let $c(x) = b(a(x))$ and let $\varvec{\beta }= (\beta _0 \ \beta _1 \ \dots \ \beta _{m-1})$ be a basis of $\mathbb F_{q^m}$ over $\mathbb F_{q}$. Let $\mathbf A\in \mathbb F_{q}^{m \times m}$, $\mathbf {C}\in \mathbb F_{q}^{m \times m}$ denote the matrix representations according to $\mathcal {B}$ of

$$\begin{aligned} \left( a(\beta _0) \ a(\beta _1) \ \dots \ a(\beta _{m-1})\right) ,\quad \left( c(\beta _0) \ c(\beta _1) \ \dots \ c(\beta _{m-1})\right) , \end{aligned}$$

respectively. Then, for the row spaces the following holds:

$$\begin{aligned} \mathcal {R}_q\left( \mathbf {C}\right) \subseteq \mathcal {R}_q\left( \mathbf {A}\right) \!. \end{aligned}$$

Proof

Consider the linearized polynomials as linear maps over $\mathbb F_{q^m}$. Then, the kernel of the map ${a}$ is equivalent to the set of roots of ${a}(x)$ in $\mathbb F_{q^m}$, considered as a vector space over $\mathbb F_{q}$. Since the roots of ${a}(x)$ are also roots of $c(x) = {b}({a}(x))$, the kernels are connected by $\ker ({a}) \subseteq \ker (c)$. For the right kernels of the matrices $\ker (\mathbf A) \subseteq \ker (\mathbf C)$ holds, and the row spaces are related by $\mathcal {R}_q\left( \mathbf C\right) \subseteq \mathcal {R}_q\left( \mathbf A\right)$. $\square$