On the Decoder Error Probability of Bounded Rank-Distance Decoders for Maximum Rank Distance Codes

TL;DR

This paper introduces elementary linear subspaces to analyze maximum rank distance (MRD) codes, demonstrating that the decoder error probability of bounded rank distance decoders decreases exponentially with the square of the error correction capability, under certain assumptions.

Contribution

It introduces the concept of elementary linear subspaces and derives new properties of MRD codes, providing an exponential bound on decoder error probability.

Findings

Decoder error probability decreases exponentially with t^2.

Elementary linear subspaces have properties similar to coordinate sets.

Error probability bound relies on errors of equal rank being equally likely.

Abstract

In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable codes. Using these properties, we show that, for MRD codes with error correction capability t, the decoder error probability of bounded rank distance decoders decreases exponentially with t^2 based on the assumption that all errors with the same rank are equally likely.