Uniquely optimal codes of low complexity are symmetric

TL;DR

This work proposes a general framework linking optimality of codes in compact metric spaces to symmetry, via the notion of unicorn spaces and unicorn sequences defined through low-complexity, uniquely optimal codes. It formalizes conjectures that uniquely optimal, computationally tractable codes should be invariant under nontrivial isometries and predicts symmetry properties under point-removal and partial-symmetry operations. The authors test these ideas across a broad spectrum of spaces, including intervals, circles, triangles, orthotopes, tori, lattices, metric graphs, ultrametric spaces, and the Hilbert cube, proving unicorn behavior in several cases and offering computational evidence for lattice-based spaces while outlining open challenges in more complex settings like the Leech lattice. The results illuminate how symmetry and optimality interact in geometric coding problems and point toward a program for verifying symmetry-driven predictions in other spaces, with potential implications for design and analysis of codes in diverse metric geometries.

Abstract

We formulate explicit predictions concerning the symmetry of optimal codes in compact metric spaces. This motivates the study of optimal codes in various spaces where these predictions can be tested.

Figures (4)

• Figure 1: Spherical codes that achieve equality in Rankin's orthoplex bound. We draw arcs between code points of geodesic distance $\pi/2$, i.e., the minimum distance of the code. (left) Six vertices of the octahedron (the three-dimensional orthoplex). This code is optimal and unique up to isometry. (middle left) Removing a vertex from the octahedron produces a size-$5$ spherical code that also achieves equality in Rankin's orthoplex bound. Such spherical codes enjoy a connected configuration space. For example, any one of the four points on the equator is able to move away from the fifth point at the north pole. This motion produces (middle right) and continues until reaching (right). Notice that for each of these codes, there exists a pair of antipodes, and the entire code is invariant under the reflection that swaps these antipodes. The characterization in Theorem \ref{['thm.orthoplex characterization']} implies that every optimal spherical code of size $5$ exhibits such a symmetry.
• Figure 2: Generic points in the configuration space of Sam Loyd's 15 puzzle. In each case, twelve of the squares are aligned with a lattice, while the other three are free to slide in their common row or column. The centers of these squares form an optimal code under the $\infty$-norm in the $3\times3$ square that contains them. This behavior is a special case of Theorem \ref{['thm.orthotope code minus 1']}.
• Figure 3: Inequivalent optimal codes of size $49$ in $\mathbb{R}^2/A_2$; i.e., there is no isometry of $\mathbb{R}^2/A_2$ that maps one code to the other. Note that the four vertices on the corners of the fundamental domain are identified as the same point; similarly, in the left-hand plot, we identify the appropriate pairs of points on parallel edges.
• Figure 4: Non-unique optimal codes in metric trees. For each $m\leq6$, consider all metric trees with nontrivial isometry group consisting of $m$ unit-length edges, and apply the linear programming bound in Subsection \ref{['sec.lp']} to find all optimal codes of size $2m$. In all but the three cases illustrated above, the optimal code is unique (and therefore invariant under the full isometry group of the metric tree). We draw a red edge between code points whose distance equals the minimum distance. In each case, the code is invariant under a nontrivial isometry of the metric tree.

Theorems & Definitions (52)

• Proposition 1
• Theorem 2
• proof
• Definition 3: informal version
• Definition 4: formal version
• Example 5
• Conjecture A
• Definition 6
• Conjecture B
• Theorem 7