On the S-matrix conjecture

Motivated with a problem in spectroscopy, Sloane and Harwit conjectured in 1976 what is the minimal Frobenius norm of the inverse of a matrix having all entries from the interval [0,1][0,1]. In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm. His proof is based on the Kiefer–Wolfowitz equivalence theorem from the approximate theory of optimal design. In this note we give a short and simple proof of his result.