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

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

💡 Research Summary

The paper investigates binary codes equipped with a poset (partially ordered set) metric, focusing on the relationship between the code’s codimension m (i.e., a code of length n and size 2^{n‑m}) and the error‑correcting radius r. The authors aim to classify all poset structures that permit a code of codimension m to be perfectly r‑error‑correcting when r is equal to m, m‑1, or m‑2.

First, the authors recall basic definitions: for a poset P on the coordinate set