# Optimal Ramp Schemes and Related Combinatorial Objects

## TL;DR

This paper introduces augmented orthogonal arrays, establishing their equivalence to ideal ramp schemes and providing new constructions that identify parameter regimes where ideal but not strong ramp schemes exist.

## Contribution

It characterizes ideal ramp schemes through augmented orthogonal arrays and offers new constructions, expanding understanding beyond strong ramp schemes.

## Key findings

- Established equivalence between ideal ramp schemes and augmented orthogonal arrays
- Provided new constructions for augmented orthogonal arrays
- Identified parameter regimes where ideal but not strong ramp schemes exist

## Abstract

In 1996, Jackson and Martin proved that a strong ideal ramp scheme is equivalent to an orthogonal array. However, there was no good characterization of ideal ramp schemes that are not strong. Here we show the equivalence of ideal ramp schemes to a new variant of orthogonal arrays that we term augmented orthogonal arrays. We give some constructions for these new kinds of arrays, and, as a consequence, we also provide parameter situations where ideal ramp schemes exist but strong ideal ramp schemes do not exist.