On irredundant orthogonal arrays

TL;DR

This paper studies irredundant orthogonal arrays (IrOAs), characterizes them via minimum Hamming distance, and constructs new families using self-dual and MDS codes, with bounds on their parameters.

Contribution

It provides a characterization of IrOAs through minimum Hamming distance and constructs new IrOAs from self-dual, MDS, and MDS-self-dual codes.

Findings

IrOAs are characterized by minimum Hamming distance at least t+1.

Either a linear code or its dual can produce an IrOA.

New families of IrOAs are constructed from special codes.

Abstract

An orthogonal array (OA), denoted by $\text{OA}(M, n, q, t)$, is an $M \times n$ matrix over an alphabet of size $q$ such that every selection of $t$ columns contains each possible $t$-tuple exactly $\lambda=M / q^t$ times. An irredundant orthogonal array (IrOA) is an OA with the additional property that, in any selection of $n - t$ columns, all resulting rows are distinct. IrOAs were first introduced by Goyeneche and \.{Z}yczkowski in 2014 to construct $t$-uniform quantum states without redundant information. Beyond their quantum applications, we focus on IrOAs as a combinatorial and coding theory problem. An OA is an IrOA if and only if its minimum Hamming distance is at least $t + 1$. Using this characterization, we demonstrate that for any linear code, either the code itself or its Euclidean dual forms a linear IrOA, giving a huge source of IrOAs. In the special case of self-dual codes, both the code and its dual yield IrOAs. Moreover, we construct new families of linear IrOAs based on self-dual, Maximum Distance Separable (MDS), and MDS-self-dual codes. Finally, we establish bounds on the minimum distance and covering radius of IrOAs.