MDS and $I$-Perfect Codes in Pomset block Metric

Abstract

In this paper, we establish the Singleton bound for pomset block codes ($(Pm,\pi )$-codes) of length $N$ over the ring ${\mathbb{Z}}_m$. We give a necessary condition for a code to be MDS in the pomset (block) metric and prove that every MDS $(Pm,\pi )$-code is an MDS $(P,\pi )$-code. Then we proceed on to find $I$-perfect and $r$-perfect codes. Further, given an ideal with partial and full counts, we look into how MDS and $I$-perfect codes relate to one another. For chain pomset, we obtain the duality theorem for pomset block codes of length $N$ over ${\mathbb{Z}}_m$; and, the weight distribution of MDS pomset block codes is then determined.

14 Pages. arXiv admin note: text overlap with arXiv:2210.15363