SummaryA necessary and sufficient condition for a balanced array of strength 2l to be a balanced fractional 2mfactorial design of resolution 2l is given. This design has the property that the main effects, two‐factor interactions,.and (l‐1)‐factor interactions are estimable ignoring the (l + 1)‐factor and higher order interactions, and that the covariance matrix of their estimates is invariant under any permutation of m factors. The above condition includes sufficient conditions given in earlier works of Shirakura (1976b