On the Existence of Retransmission Permutation Arrays

TL;DR

This paper proves the existence of retransmission permutation arrays with specific corner coverage properties for all sizes, including Latin variants, which are useful in overlapping channel transmission applications.

Contribution

It establishes the existence of all four types of RPAs and Latin RPAs for every positive integer size, filling a theoretical gap in combinatorial design theory.

Findings

Existence of type-1,2,3,4 RPAs for all n

Existence of type-1,2 Latin RPAs for all n

Theoretical foundation for applications in overlapping channels

Abstract

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of ${1, ..., n}$, and for $1\leq i\leq n$, all $n$ symbols occur in each $i\times\lceil\frac{n}{i}\rceil$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-$1,2,3,4$ $\RPA(n)$ and a type-1,2 latin $\RPA(n)$.