Optimal and minimal linear codes from generalized hierarchical posets
Optimal and minimal $p$-ary linear codes from generalized order ideals of hierarchical posets
Information Theory
Summary
The paper extends previous work on special binary error-correcting codes built using ordered structures called posets to codes over larger alphabets (p-ary codes). The authors define new types of order ideals by attaching multiplicities to poset elements, resulting in novel code constructions that differ from known classical methods. They calculate important properties like weight distributions and identify families of codes that are both optimal and minimal, some breaking previously known conditions. This contributes to the theory and design of efficient error-correcting codes with specific minimality properties.
What this means in practice
- •For communication engineers: Design error-correcting codes with optimized parameters for data transmission systems using hierarchical poset-based structures.
- •For coding theory developers: Generate new minimal p-ary codes that defy classical minimality conditions for specialized cryptographic and coding applications.
Authors
Rumi Melih Pelen
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.