A Generalization of Lee Codes

Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb-Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler.

💡 Research Summary

The paper introduces a new combinatorial object called a Perfect Distance‑Dominating Set (PDDS) motivated by a practical problem in computer architecture—namely, the need to assign resources (memory addresses, network routes, etc.) such that every node is “dominated” by a center within a prescribed distance. Formally, given a graph (G=(V,E)) and an integer (d\ge 1), a set (S\subseteq V) is a PDDS if the vertices of (V) can be partitioned into disjoint balls of radius (d) centred at the vertices of (S); each non‑center vertex must be at distance at most (d) from exactly one centre. This definition simultaneously generalises three well‑studied families: perfect Lee codes (the case of the integer lattice (\mathbb Z^n) with Lee distance), diameter‑perfect codes (where the graph diameter replaces the Lee metric), and ordinary domination sets (the special case (d=1)).

The authors first motivate the definition through concrete architectural scenarios, then develop a systematic theory of PDDS. The main contributions can be grouped into four themes.

1. Formalisation and Relation to Existing Codes – The paper precisely defines PDDS, introduces terminology (balls, centres, domination radius) and shows how perfect Lee codes, diameter‑perfect codes, and classic domination sets appear as specialisations. A concise diagram summarises these inclusions, making clear that PDDS is the natural umbrella concept.

2. Construction of Lattice‑Like PDDS – The most important constructive results concern “lattice‑like” PDDS, i.e., sets whose centres form a sub‑lattice of (\mathbb Z^n) (or an analogous periodic structure in other graphs). For (\mathbb Z^2) the authors recover the classical tiling by ((2r+1)\times(2r+1)) squares, proving that the centres placed at ((2r+1)\mathbb Z\times(2r+1)\mathbb Z) yield a PDDS of radius (r). In higher dimensions they introduce a Coset‑Reduce technique: choose integers (a_1,\dots,a_n) related to (d) and tile (\mathbb Z^n) by the sub‑lattice (a_1\mathbb Z\times\cdots\times a_n\mathbb Z); each tile contains a single (d)‑ball, guaranteeing the PDDS property. For hypercubes (Q_n) they construct PDDS by selecting centres among vertices of a fixed Hamming weight and using bit‑flip neighbourhoods. All these constructions are explicit, algorithmic and, crucially, lead to constant‑time decoding: given a vertex, one merely reduces its coordinates modulo the lattice periods to locate the unique centre.

3. Non‑Existence Results – Not every pair ((n,d)) admits a PDDS. The authors employ two complementary arguments. The first is a counting (density) argument: the size of a radius‑(d) ball, denoted (B(d)), must divide the total number of vertices in any finite quotient of the graph. When this divisibility fails (e.g., (\mathbb Z^3) with (d=2) where (B(2)=125) does not tile any rectangular fundamental domain), a periodic PDDS cannot exist. The second argument uses algebraic topology and group‑theoretic properties of the underlying lattice: if the lattice periods share a non‑trivial common factor with (2d+1), the induced coset structure prevents a perfect tiling. These non‑existence proofs extend known impossibility results for perfect Lee codes to the broader PDDS framework.

4. A PDDS Version of the Golomb‑Welch Conjecture – The classic Golomb‑Welch conjecture (1970) asserts that perfect Lee codes exist only for the trivial cases ((n, r)=(1, \text{any})) or ((2,1)). The paper reformulates this conjecture in PDDS language: a lattice‑like PDDS of radius (d) in (\mathbb Z^n) can exist only when ((2d+1)^n) divides the volume of the fundamental parallelepiped of the centre lattice. This yields a number‑theoretic condition that generalises the original conjecture. The authors verify the condition for several low‑dimensional instances (e.g., all ((n, d)) with (n=2) or (d=1)) and prove that for (n\ge 3) and (d\ge 2) the condition cannot be satisfied, thereby providing a PDDS‑based partial proof of the Golomb‑Welch conjecture.

Beyond the theoretical contributions, the paper discusses practical implications. Lattice‑like PDDS allow hardware designers to implement address translation or routing tables using simple modular arithmetic, dramatically reducing lookup latency and memory overhead. The authors sketch a prototype mapping scheme for a many‑core processor where each core’s address space is a PDDS ball, showing how the scheme guarantees bounded hop distance and conflict‑free allocation.

In summary, the work establishes PDDS as a unifying abstraction for a variety of coding and domination problems, supplies explicit constructions for a wide class of graphs, proves rigorous non‑existence results, and extends a celebrated open problem (the Golomb‑Welch conjecture) into this broader setting. The lattice‑like nature of the constructed PDDS makes the theory immediately applicable to real‑world system design, while the open questions left—especially concerning non‑lattice PDDS and higher‑dimensional density bounds—provide a fertile ground for future research at the intersection of combinatorial coding theory, graph theory, and computer architecture.