The State of Lexicodes and Ferrers Diagram Rank-Metric Codes

• Main Author: Antrobus, Jared E.
• Format: Others
• Published: UKnowledge 2019
• Subjects: coding theory; lexicode; Ferrers diagram; rank-metric; MRD code; Algebra
• Online Access: https://uknowledge.uky.edu/math_etds/66; https://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1067&context=math_etds

In coding theory we wish to find as many codewords as possible, while simultaneously maintaining high distance between codewords to ease the detection and correction of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. In this work I describe the recent advances of this problem in the contexts of lexicodes and Ferrers diagram rank-metric codes. In the first chapter, we study lexicodes. For a ring R, we describe a lexicographic ordering of the left R-module Rn. With this ordering we set up a greedy algorithm which sequentially selects vectors for which all linear combinations satisfy a given property. The resulting output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. We describe a generalization of the algorithm to finite principal ideal rings. In the second chapter, we investigate Ferrers diagram rank-metric codes, which play a role in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. We describe several solved cases of the conjecture, and further contribute new ones. In addition, probabilities for maximal Ferrers diagram codes and MRD codes are investigated in a new light. It is shown that for growing field size, the limiting probability depends highly on the Ferrers diagram.