The poset metrics that allow binary codes of codimension m to be m-, (m-1)-, or (m-2)-perfect

TL;DR

This paper characterizes all poset metrics that enable binary codes of a given codimension to be perfect or nearly perfect, providing conditions for their existence and nonexistence, with specific examples involving crown posets and disjoint chains.

Contribution

It describes all poset metrics allowing codes of a fixed codimension to be perfect or near-perfect and establishes conditions for the nonexistence of such codes in certain poset structures.

Findings

Characterization of all poset metrics for perfect codes of given codimension

Conditions for the nonexistence of perfect poset codes in specific structures

Proof of nonexistence of R-perfect codes for crown and disjoint chain posets

Abstract

A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of R-perfect poset codes for some R in the case of the crown poset and in the case of the union of disjoin chains. Index terms: perfect codes, poset codes