An Improvement on Ranks of Explicit Tensors

Abstract: We give constructions of n^k x n^k x n tensors of rank at least 2n^k - O(n^k-1). As a corollary we obtain an [n]^r shaped tensor with rank at least 2n^r/2 - O(n^r/2-1) when r is odd. The tensors are constructed from a simple recursive pattern, and the lower bounds are proven using a partitioning theorem developed by Brockett and Dobkin. These two bounds are improvements over the previous best-known explicit tensors that had ranks n^k and n^r/2 respectively