Maximal Separation on \(2-D\) Arrays

Abstract

Given \(m, n\) and \(2 \leq l \leq mn\), we study the problem of separating \(l\) symbols on an \(m \times n\) array such that the minimum \(\ell_1\) distance between any two of the \(l\) symbols is as large as possible. This problem is similar in nature to the well-known Tammes’ problem where one tries to achieve the largest angular separation for a given number of points on a \(2-D\) or higher dimensional sphere. It is also closely related to the well-studied problem of constructing optimal interleaving schemes for correcting error bursts in multi-dimensional digital data where a burst can be an arbitrarily shaped connected region in the array. Moreover, the interest in studying this problem also arises from considerations of minimizing the risk of multiple nearby node failures in a distributed data storage system (or a similar industrial network) in the event of a relatively large scale random disruption. We derive bounds on the maximum possible distance of separation for general \(m,n\) and \(l\), and provide also optimal constructions in several special cases including small and large \(l\) values, small \(m\) (or \(n\)) values, and \(n-1 \geq (l-1)(m-1)\).