Weighted Coordinates Poset Block Codes
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Given [n]={1,2,…,n}, a partial order ≼ on [n], a label map π : [n] →ℕ defined by π(i) = k_i with ∑_i=1^nπ (i) = N, the direct sum 𝔽_q^k_1⊕𝔽_q^k_2⊕…⊕𝔽_q^k_n of 𝔽_q^N, and a weight function w on 𝔽_q, we define a poset block metric d_(P,w,π) on 𝔽_q^N based on the poset P=([n],≼). The metric d_(P,w,π) is said to be weighted coordinates poset block metric ((P,w,π)-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,π)-metric) introduced by M. M. S. Alves et al. We determine the complete weight distribution of a (P,w,π)-space, thereby obtaining it for (P,w)-space, (P,π)-space, π-space, and P-space as special cases. We obtain the Singleton bound for (P,w,π)-codes and for (P,w)-codes as well. In particular, we re-obtain the Singleton bound for any code with respect to (P,π)-metric and P-metric. Moreover, packing radius and Singleton bound for NRT block codes are found.
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