Computing coset leaders and leader codewords of binary codes

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 $\mathrm{C}\mathrm{L}(\mathcal{C})$ 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.