Orphans of the first order Reed-Muller codes

If C is a code, an orphan is a coset that is not a descendant. Orphans arise naturally in the investigation of the covering radius. Case C has only even-weight vectors and minimum distance of at least four. Cosets that are orphans are characterized, and then the existence is proved of a family of orphans of first-order Reed-Muller codes R(1, m). For m>