In 1988, Sakata extended the Berlekamp – Massey (BM) algorithm from 1 dimension to n dimensions. The so-called Berlekamp – Massey – Sakata (BMS) algorithm can be used on a table for finding a Gröbner basis of a 0-dimensional ideal of relations satisfied by said table. We investigate this problem using linear algebra techniques, such as what have been done for the BM algorithm by Kaltofen and Yuhasz, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity. We first recall the definition of multidimensional linear recurrent sequences for 0-dimensional ideals given by Sakata or Fitzpatrick and Norton. We design several algorithms for computing a Gröbner basis of the ideal of relations. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. Under genericity assumptions, this probabilistic technique essentially reduces the problem to using the efficient 1-dimensional BM algorithm. Each probe to the table can be expensive. We thus consider the table in the black-box model and look for minimizing the number of probes to the table in our complexity model. We thus design a second algorithm, Scalar-FGLM, requiring, in general, less probes to the table. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make this