The Zero Cubes Free and Cubes Unique Multidimensional Constraints

TL;DR

This paper investigates two multidimensional array constraints related to zero cubes and unique cubes, providing conditions for asymptotic rates, redundancy bounds, and efficient encoding algorithms, extending to arbitrary box constraints.

Contribution

It introduces new conditions and encoding algorithms for multidimensional array constraints involving zero cubes and unique cubes, with extensions to arbitrary box volumes.

Findings

Asymptotic rate approaches 1 for certain cube sizes.

Redundancy can be at most one symbol under specific conditions.

Efficient encoding algorithms are developed for both constraints.

Abstract

This paper studies two families of constraints for two-dimensional and multidimensional arrays. The first family requires that a multidimensional array will not contain a cube of zeros of some fixed size and the second constraint imposes that there will not be two identical cubes of a given size in the array. These constraints are natural extensions of their one-dimensional counterpart that have been rigorously studied recently. For both of these constraint we present conditions of the size of the cube for which the asymptotic rate of the set of valid arrays approaches 1 as well as conditions for the redundancy to be at most a single symbol. For the first family we present an efficient encoding algorithm that uses a single symbol to encode arbitrary information into a valid array and for the second family we present a similar encoder for the two-dimensional case. The results in the paper are also extended to similar constraints where the sub-array is not necessarily a cube, but a box of arbitrary dimensions and only its volume is bounded.