Bounds on Covering Codes with the Rank Metric
Maximilien Gadouleau, Zhiyuan Yan
TL;DR
This paper explores the geometric properties of the rank metric space and derives bounds on the size of codes needed to cover the space within a certain radius, aiding the design and analysis of rank metric codes.
Contribution
It provides analytical expressions for intersections of rank metric balls and establishes bounds on code cardinality based on these geometric insights.
Findings
Derived an analytical expression for the intersection of two rank metric balls
Established upper bounds on the volume of unions of rank metric balls
Provided bounds on the minimum size of rank covering codes
Abstract
In this paper, we investigate geometrical properties of the rank metric space and covering properties of rank metric codes. We first establish an analytical expression for the intersection of two balls with rank radii, and then derive an upper bound on the volume of the union of multiple balls with rank radii. Using these geometrical properties, we derive both upper and lower bounds on the minimum cardinality of a code with a given rank covering radius. The geometrical properties and bounds proposed in this paper are significant to the design, decoding, and performance analysis of rank metric codes.
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Taxonomy
TopicsCooperative Communication and Network Coding · Coding theory and cryptography · DNA and Biological Computing