Lexicodes over Rings

In this paper, we consider the construction of linear lexicodes over finite chain rings by using a $B$-ordering over these rings and a selection criterion. % and a greedy Algorithm. As examples we give lexicodes over $\mathbb{Z}_4$ and $\mathbb{F}_2+u\mathbb{F}_2$. %First, greedy algorithms are presented to construct %lexicodes using a multiplicative property. Then, greedy algorithms %are given for the case when the selection criteria is not %multiplicative such as the minimum distance constraint. It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.