  In this paper we use the Gr\"obner representation of a binary linear code
$\mathcal C$ to give efficient algorithms for computing the whole set of coset
leaders, denoted by $\mathrm{CL}(\mathcal C)$ and the set of leader codewords,
denoted by $\mathrm L(\mathcal C)$. The first algorithm could be adapted to
provide not only the Newton and the covering radius of $\mathcal C$ but also to
determine the coset leader weight distribution. Moreover, providing the set of
leader codewords we have a test-set for decoding by a gradient-like decoding
algorithm. Another contribution of this article is the relation stablished
between zero neighbours and leader codewords.
