On irredundant orthogonal arrays

An orthogonal array (OA), denoted by OA ( M , n , q , t ) , is an M × n matrix over an alphabet of size q such that every selection of t columns contains each possible t -tuple exactly λ = 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 Życzkowski 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 particular, the self-dual codes 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.