Constructing de Bruijn sequences with co-lexicographic order: The k-ary Grandmama sequence

A k-ary de Bruijn sequence of order n is a circular k-ary string of length kⁿ which contains every k-ary string of length n exactly once as a substring. It is well-known that a k-ary de Bruijn sequence of order n can be constructed by concatenating the aperiodic prefixes of the k-ary necklaces of length n in lexicographic order. In this article we prove that an alternate de Bruijn sequence is created by replacing lexicographic order with co-lexicographic order. We also provide a simple successor rule for generating each successive symbol in O(n)-time.