Optimal minimal Linear codes from posets

TL;DR

This paper extends the construction of minimal and optimal binary linear codes from simplicial complexes to arbitrary posets, specifically hierarchical two-level posets, and determines their weight distributions.

Contribution

It introduces new methods for constructing binary linear codes from hierarchical posets and analyzes their weight distributions, leading to new optimal and minimal codes.

Findings

Constructed binary linear codes from hierarchical posets with two levels.

Determined the weight distributions of these codes.

Identified new optimal and minimal codes not satisfying Ashikhmin-Barg condition.

Abstract

Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun {\em et al.} We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the condition of Ashikhmin-Barg.