MDS codes based on orthogonality of quasigroups.

In this paper, we propose a novel method for constructing maximum distance separable (MDS) codes based on the extended invertibility and orthogonality of quasigroups. We provide various methods of constructing an orthogonal system of <italic>k</italic>-ary operations over Q2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q^2$$\end{document} using a special type of <italic>k</italic>-ary operations over <italic>Q</italic>, where <italic>Q</italic> is any arbitrary finite set. Then we use concepts of strong orthogonality of <italic>k</italic>-ary operations to establish a connection between orthogonality and linear recursive MDS codes. We illustrate these new classes of MDS codes using the proposed techniques and enumerate such codes using SageMath. [ABSTRACT FROM AUTHOR]