Constructions of Augmented Orthogonal Arrays

TL;DR

This paper characterizes augmented orthogonal arrays (AOAs) in terms of orthogonal arrays (OAs) and MDS codes, providing new constructions and infinite classes of AOAs.

Contribution

It establishes a precise equivalence between AOAs, OAs, and MDS codes, and introduces new constructions and infinite classes of AOAs.

Findings

AOA$(s,t,k,v)$ exists iff OA$(t,k,v)$ can be partitioned into $v^{t-s}$ OA$(s,k,v)$

Linear AOA$(s,t,k,q)$ exists iff a linear MDS code contains a linear MDS subcode

New constructions and infinite classes of AOAs are provided

Abstract

Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and augmented orthogonal arrays (Discrete Math. 341 (2018), 299-307). In this paper, we show that there is an AOA$(s,t,k,v)$ if and only if there is an OA$(t,k,v)$ which can be partitioned into $v^{t-s}$ subarrays, each being an OA$(s,k,v)$, and that there is a linear AOA$(s,t,k,q)$ if and only if there is a linear maximum distance separable (MDS) code of length $k$ and dimension $t$ over $\mathbb{F}_q$ which contains a linear MDS subcode of length $k$ and dimension $s$ over $\mathbb{F}_q$. Some constructions for AOAs and some new infinite classes of AOAs are also given.