Shorter strings containing all k‐element permutations

We consider the problem of finding short strings that contain all permutations of order k over an alphabet of size n, with k≤n. We show constructively that k(n-2)+3 is an upper bound on the length of shortest such strings, for n≥k≥10. Consequently, for n≥10, the shortest strings that contain all permutations of order n have length at most n²-2n+3. These two new upper bounds improve with one unit the previous known upper bounds.