Optimal and minimal $p$-ary linear codes from generalized order ideals of hierarchical posets

Source abstract

Hyun, Kim, Wu and Yue constructed optimal and minimal binary linear codes from order ideals of hierarchical posets with two levels. Two different generalizations of the underlying antichain (simplicial complex) setting to odd characteristic are known: down-sets of $\mathbb{F}_p^n$ under the componentwise order, and support-closed subsets of $\mathbb{F}_q^m$. No generalization of the poset setting itself has appeared. We introduce generalized order ideals of a poset of order $p-1$, obtained by attaching multiplicities in ${0,\dots,p-1}$ to the elements of a poset, and study the two natural notions of order ideal that arise for hierarchical posets with two levels. Whenever the ideal meets the upper level, the resulting defining sets are neither down-sets nor support-closed. We determine the weight distributions of the associated complement codes, exhibit a family of Griesmer codes in which the upper element carries an arbitrary multiplicity, and, via the characteristic function of a generalized order ideal, obtain an infinite family of minimal $p$-ary codes of length $p^n-1$ and dimension $n+1$ violating the Ashikhmin--Barg condition.