# A Note on the Injection Distance

## TL;DR

This paper demonstrates that Gabidulin codes are asymptotically optimal under the injection distance in noncoherent network coding, extending their known optimality from subspace distance.

## Contribution

It establishes the asymptotic optimality of Gabidulin codes with respect to the injection distance in noncoherent network coding.

## Key findings

- Gabidulin codes are asymptotically optimal for injection distance.
- Extension of optimality from subspace distance to injection distance.
- Supports the effectiveness of these codes in noncoherent network coding scenarios.

## Abstract

Koetter and Kschischang showed in [R. Koetter and F.R. Kschischang, "Coding for Errors and Erasures in Random Network Coding," IEEE Trans. Inform. Theory, {54(8), 2008] that the network coding counterpart of Gabidulin codes performs asymptotically optimal with respect to the subspace distance. Recently, Silva and Kschischang introduced in [D. Silva and F.R. Kschischang, "On Metrics for Error Correction in Network Coding," To appear in IEEE Trans. Inform. Theory, ArXiv: 0805.3824v4[cs.IT], 2009] the injection distance to give a detailed picture of what happens in noncoherent network coding. We show that the above codes are also asymptotically optimal with respect to this distance.