Optimal Ramp Schemes and Related Combinatorial Objects
This work solves a theoretical gap in combinatorial cryptography for researchers in secret sharing and coding theory, though it is incremental as it builds on prior equivalence results.
The paper addresses the lack of characterization for ideal ramp schemes that are not strong, showing their equivalence to augmented orthogonal arrays and providing constructions that demonstrate existence in parameter situations where strong ideal ramp schemes do not exist.
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.