Reconstruction of hypermatrices from subhypermatrices

Abstract

For a given n, what is the smallest number k such that every sequence of length n is determined by the multiset of all its k-subsequences? This is called the k-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of $n\times n$-matrices from submatrices. Previous works show that the smallest k is at most $O\left(n^{\frac{1}{2}}\right)$ for sequences and at most $O\left(n^{\frac{2}{3}}\right)$ for matrices. We study this k-deck problem for general dimension d and prove that, the smallest k is at most $O\left(n^{\frac{d}{d+1}}\right)$ for reconstructing any d dimensional hypermatrix of order n from the multiset of all its subhypermatrices of order k.