2025/01/21 by Shriwastva, Atul Kumar, Selvaraj, R. S.
#06A06 #15A03 #94B05 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2501.11978
In this paper, we determine the complete weight distribution of the space \mathbbFqN endowed by the weighted coordinates poset block metric ((P,w,π)-metric), also known as the (P,w,π)-space, thereby obtaining it for (P,w)-space, (P,π)-space, π-space, and P-space as special cases. Further, when P is a chain, the resulting space is called as Niederreiter-Rosenbloom-Tsfasman (NRT) weighted block space and when P is hierarchical, the resulting space is called as weighted coordinates hierarchical poset block space. The complete weight distribution of both the spaces are deduced from the main result. Moreover, we define an I-ball for an ideal I in P and study the characteristics of it in (P,w,π)-space. We investigate the relationship between the I-perfect codes and t-perfect codes in (P,w,π)-space. Given an ideal I, we investigate how the maximum distance separability (MDS) is related with I-perfect codes and t-perfect codes in (P,w,π)-space. Duality theorem is derived for an MDS (P,w,π)-code when all the blocks are of same length. Finally, the distribution of codewords among r-balls is analyzed in the case of chain poset, when all the blocks are of same length.