Reconstructing words using queries on subwords or factors

We study word reconstruction problems. Improving a previous result by P. Fleischmann, M. Lejeune, F. Manea, D. Nowotka and M. Rigo, we prove that, for any unknown word $w$ of length $n$ over an alphabet of cardinality $k$, $w$ can be reconstructed from the number of occurrences as subwords (or scattered factors) of $O(k^2\sqrt{n{\log }_2(n)})$ words. Two previous upper bounds obtained by S. S. Skiena and G. Sundaram are also slightly improved: one when considering information on the existence of subwords instead of on the numbers of their occurrences, and, the other when considering information on the existence of factors.