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 poset‑metric codes of codimension M (i.e., codes of length N with 2^{N‑M} codewords) and determines precisely which poset structures admit M‑perfect, (M‑1)‑perfect, or (M‑2)‑perfect codes. A poset metric d_P is defined by the size of the upward‑closed ideal generated by the support of the difference of two vectors. The authors first recall that a code is R‑perfect if the balls of radius R under d_P partition the whole space ℱ₂^N. For a code of codimension M the maximum correctable error number is M, so the most natural case is M‑perfectness.

The main contribution is a set of necessary and sufficient conditions on the underlying poset P for each of the three levels of perfection. For M‑perfectness the poset must be “M‑uniform”: every minimal element of P generates an ideal of exactly M coordinates. Under this uniformity the radius‑M balls are disjoint and cover ℱ₂^N, yielding an M‑perfect code. Conversely, if any minimal element generates an ideal of size ≠ M, overlap or gaps appear and perfectness fails.

For (M‑1)‑perfect and (M‑2)‑perfect codes the situation is more intricate. The authors introduce a chain‑decomposition of P (expressing P as a disjoint union of chains) and define a chain‑length vector (ℓ₁,…,ℓ_k). They prove that (M‑1)‑perfectness is possible only when the set of minimal elements can be split into exactly two antichains, each forming a chain of lengths ℓ₁ and ℓ₂ with ℓ₁+ℓ₂ = M‑1. For (M‑2)‑perfectness a three‑chain decomposition (or more) is allowed, but the lengths must satisfy a system of linear inequalities derived from the volume formula of poset balls. These conditions are both necessary and sufficient, and they lead to an explicit construction method for such codes.

A substantial part of the paper is devoted to non‑existence results. By comparing the height h(P) (maximum chain length) and the width w(P) (size of a maximum antichain), the authors establish a general inequality h·w < M·(M‑1)/2 that guarantees the impossibility of any M‑perfect code. This inequality captures the intuition that a poset that is too “thin” or too “tall” cannot produce balls that tile the space without overlap.

The authors apply the general theory to two concrete families of posets. First, the crown poset C_n (a cycle of n elements with the natural partial order) is shown to admit an M‑perfect code only when the radius equals n‑1; for smaller radii the inequality above rules out perfectness. Second, they consider the disjoint union of k chains of equal length ℓ. By plugging the parameters into the height‑width condition they prove that for many (k,ℓ) pairs no (M‑1)‑ or (M‑2)‑perfect codes exist, thereby illustrating the strength of the derived criteria.

The paper concludes by discussing implications for coding theory. Poset metrics generalize the Hamming metric and allow the design of codes that are perfectly adapted to hierarchical or network‑structured data. The precise characterization of posets that support perfect codes opens the way to construct new families of optimal codes in settings where Hamming‑perfect codes are impossible. Open problems include extending the results to non‑binary alphabets, exploring the existence of non‑linear perfect poset codes, and developing efficient decoding algorithms that exploit the underlying poset structure.