The so-called Berlekamp~-- Massey~-- Sakata algorithm computes a Gröbner basis of a $0$-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp~-- Massey algorithm to $n$-dimensional tables, for $n>1$. We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This \textsc{FGLM}-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a \emph{multi-Hankel} matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the \emph{geometry} of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode $n$-cyclic codes, a generalization in dimension $n$ of Reed Solomon codes. We show that the multi-Hankel matrices are heavily structured when using the \textsc{LEX} ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.