Weighted Coordinates Poset Block Codes

Given $[n]=\{1,2,\dots ,n\}$, a partial order $⪯$ on $[n]$, a label map $\pi :[n]\to \mathbb{N}$ defined by $\pi (i)=k_i$ with ${\sum }_{i=1}^n\pi (i)=N$, the direct sum ${\mathbb{F}}_q^{k_1}\oplus {\mathbb{F}}_q^{k_2}\oplus \dots \oplus {\mathbb{F}}_q^{k_n}$ of ${\mathbb{F}}_q^N$, and a weight function $w$ on ${\mathbb{F}}_q$, we define a poset block metric $d_{(P,w,\pi )}$ on ${\mathbb{F}}_q^N$ based on the poset $P=([n],⪯)$. The metric $d_{(P,w,\pi )}$ is said to be weighted coordinates poset block metric ($(P,w,\pi )$-metric). It extends the weighted coordinates poset metric ($(P,w)$-metric) introduced by L. Panek and J. A. Pinheiro and generalizes the poset block metric ($(P,\pi )$-metric) introduced by M. M. S. Alves et al. We determine the complete weight distribution of a $(P,w,\pi )$-space, thereby obtaining it for $(P,w)$-space, $(P,\pi )$-space, $\pi$-space, and $P$-space as special cases. We obtain the Singleton bound for $(P,w,\pi )$-codes and for $(P,w)$-codes as well. In particular, we re-obtain the Singleton bound for any code with respect to $(P,\pi )$-metric and $P$-metric. Moreover, packing radius and Singleton bound for NRT block codes are found.