MDS codes based on orthogonality of quasigroups

Satish Kumar, Harshdeep Singh, Indivar Gupta, Ashok Ji Gupta

Abstract: 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 k-ary operations over \(Q^2\) using a special type of k-ary operations over Q, where Q is any arbitrary finite set. Then we use concepts of strong orthogonality of k-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.