Let P be a partial order on [n] = {1,2,...,n}, Fqnbe the linear space of n-tuples over a finite field Fqand w be a weight on Fq. In this paper, we consider metrics on \mathbb Fqninduced by chain orders P over [n] and weights w over \mathbb Fq, and we determine the cardinality of all optimal anticodes and completely classify them. Moreover, we determine all diameter perfect codes for a set of relevant instances on the aforementioned metric spaces.

Let P = ([n], ≤P) be a poset on [n] = {1, 2,⋯, n}, Fqnbe the linear space of n-tuples over a finite field Fq and w be a weight on Fq. In this paper we consider metrics on Fqnwhich are induced by posets over [n] and weights over Fq. Such family of metrics extend both the additive metric induced by the weight w (when the poset is an anti-chain) and the poset metrics (when the weight is the Hamming weight). Furthermore, the pomset metrics is also a particular case of our construction, consequently, we provide a simpler approach to these metrics without using the multiset structure originally proposed. For the general case, we provide a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we (re)obtain the groups of linear isometries of the poset, pomset and additive metric spaces. When considering a chain order, we develop, for codes on these spaces, several of the invariants and properties found in the classical coding theory. Our construction of metrics based on partial orders and any weight over the base field, highlights the dependence of the poset metric over Fqnwith the Hamming metric on Fqand the additive property of its extension on Fqn.

Poset and block metrics were introduced in recent years as alternative metrics to study error correcting codes. Poset-block codes were introduced in 2008, encompassing both poset and block metrics. In this paper, we study a family of such metrics, the Niederreiter-Rosenbloom-Tsfasman block metrics. In this context, we classify the classes of equivalent codes, describe canonical representatives of each class and develop much of the classical theory of error correcting codes for Niederreiter-Rosenbloom-Tsfasman block codes, including determination of packing radius and classification of MDS and perfect codes, determination of covering radius and characterization of quasi-perfect codes, and the description of an algorithm for syndrome decoding.