Local unitary equivalence of orthogonal arrays and related linear codes

Abstract

Orthogonal arrays (OAs) are combinatorial configurations with applications in experimental design, error-correcting codes, and quantum information. Local unitary (LU) equivalence provides a natural framework for classifying multipartite entangled states. Using the correspondence between OAs and quantum states, Goyeneche and Życzkowski [Phys. Rev. A, 2014, 90: 022316] posed the problem of determining when OAs are $LU$ equivalent. In this paper, we construct OAs from generator matrices and establish conditions for Fourier-based $LU$ equivalence of OAs and irredundant orthogonal arrays (IrOAs). For prime alphabets, we identify the Fourier partners of the linear OAs considered here with the arrays of their corresponding dual codes. This gives an explicit connection between $LU$ equivalence and coding theory. We also identify families of $LU$ equivalent OAs arising from specific classes of linear codes.