Improved decoding of symmetric rank metric errors

TL;DR

This paper advances decoding techniques for symmetric rank metric errors, demonstrating new decodable code families at rates below 1/2 and proposing a decoder for higher rates, enhancing error correction capabilities.

Contribution

It introduces new decodable code families for symmetric errors at rates below 1/2 and a decoder for Gabidulin codes at rates above 1/2, improving error correction in rank metric codes.

Findings

Decodable code families exist for rates < 1/2 correcting symmetric errors.

A deterministic decoder for Gabidulin codes corrects symmetric errors up to rank n-k.

The decoders are worst-case and applicable to symmetric error patterns.

Abstract

We consider the decoding of rank metric codes assuming the error matrix is symmetric. We prove two results. First, for rates $<1/2$ there exists a broad family of rank metric codes for which any symmetric error pattern, even of maximal rank can be corrected. Moreover, the corresponding family of decodable codes includes Gabidulin codes of rate $<1/2$. Second, for rates $>1/2$, we propose a decoder for Gabidulin codes correcting symmetric errors of rank up to $n-k$. The two mentioned decoders are deterministic and worst case.