Nonexistence of a few binary orthogonal arrays

TL;DR

This paper uses combinatorial algorithms to prove the nonexistence of certain binary orthogonal arrays and to narrow down feasible distributions, advancing understanding of their existence and classification.

Contribution

It introduces a new combinatorial approach that conclusively proves the nonexistence of specific binary orthogonal arrays and reduces feasible distributions for existing arrays.

Findings

Proved nonexistence of arrays with parameters (9,96,4), (10,96,5), (10,112,4), (11,112,5), (11,112,4), (12,112,5)

Resolved first undecided cases of array existence

Reduced feasible distributions for existing arrays, aiding classification

Abstract

We develop and apply combinatorial algorithms for investigation of the feasible distance distributions of binary orthogonal arrays with respect to a point of the ambient binary Hamming space utilizing constraints imposed from the relations between the distance distributions of connected arrays. This turns out to be strong enough and we prove the nonexistence of binary orthogonal arrays of parameters (length, cardinality, strength)$\ =(9,6.2^4=96,4)$, $(10,6.2^5,5)$, $(10,7.2^4=112,4)$, $(11,7.2^5,5)$, $(11,7.2^4,4)$ and $(12,7.2^5,5)$, resolving the first cases where the existence was undecided so far. For the existing arrays our approach allows substantial reduction of the number of feasible distance distributions which could be helpful for classification results (uniqueness, for example).