Abstract: Let $\mathcal C$ be an $[n,k,d]-$linear code with generating matrix $A$; this is assumed to be a rank $k$, $k\times n$ matrix with entries in a field $\mathbb K$. Computing $d$, the minimum distance, is in general an NP-hard problem and finding a good lower bound has been a major question in algebraic coding theory. If the matrix $A$ has no proportional nor zero columns, consider $\Gamma\subset\mathbb P^{k-1}$ the set of $n$ distinct points with homogeneous coordinates the entries of each column of $A$. In this talk we present a good lower bound for $d$ in terms of a certain shift in the last free module of the graded minimal free resolution of $\mathbb K$ $[x_1,\ldots,x_k]/I(\Gamma)$. We also present the De Boer-Pellikaan method to compute $d$. As a consequence of this method one can obtain a symbolic description of the variety of minimal codewords in terms of the top $Ext$ module of a certain ideal generated by products of linear forms.