Some Gabidulin Codes cannot be List Decoded Efficiently at any Radius

TL;DR

This paper proves that many Gabidulin codes cannot be efficiently list decoded beyond half their minimum distance, establishing exponential list sizes and extending results to related subspace codes.

Contribution

It introduces new bounds on list sizes for Gabidulin codes, showing they cannot be efficiently decoded beyond certain radii, a novel result for linear rank-metric codes.

Findings

Exponential list sizes exist for certain Gabidulin codes.

Efficient list decoding beyond half the minimum distance is impossible for these codes.

Results extend to constant dimension subspace codes via lifting.

Abstract

Gabidulin codes can be seen as the rank-metric equivalent of Reed-Solomon codes. It was recently proven, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially-sized lists exist for codes with specific parameters. The second bound presents exponentially-sized lists explicitly, for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes.