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×n-matrices from submatrices. Previous works show that the smallest k is at most O(n12) for sequences and at most O(n23) for matrices. We study this k-deck problem for general dimension d and prove that, the smallest k is at most O(ndd+1) for reconstructing any d dimensional hypermatrix of order n from the multiset of all its subhypermatrices of order k.

中文翻译：

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从子超矩阵重建超矩阵


对于给定的 n，最小的数字 k 是多少，使得每个长度为 n 的序列都由其所有 k 子序列的多集确定？这称为序列重建的 k 甲板问题，并已推广到二维情况 – 从子矩阵重建 n×n 矩阵。以前的工作表明，对于序列，最小的 k 最大为 O（n12），对于矩阵最大为 O（n23）。我们针对一般维度 d 研究这个 k-deck 问题，并证明，最小的 k 最多为 O（ndd+1），用于从其所有 k 阶超矩阵的多重集重建任何 n 阶的 d 维超矩阵。