On the number of ideals of the Niederreiter–Rosenbloom–Tsfasman poset and lower bounds on covering codes

The Niederreiter-Rosenbloom-Tsfasman (NRT) partially ordered set (poset) has been widely investigated due to the seminal papers on poset metrics of Brualdi et al. (Discrete Math 147:52–75, 1995) and on the m-metric by Rosenbloom and Tsfasman (Probl Inform Trans 33:45–52, 1997). In this work, we investigate new combinatorial identities on the number of ideals of cardinality i of the NRT poset having exactly j maximal elements. These identities are applied to evaluate the cardinality of a sphere of radius R over the NRT metric. A combination of previous results enables us to generalize Hämäläinen’s method for covering codes over the NRT metric, which improves the sphere covering bound for many instances. A table shows several numerical improvements of the sphere covering bound by using Hämäläinen’s method. As a consequence, we obtain a criterion for the non-existence of perfect codes in NRT spaces.