The Combinatorial Rank of Subsets: Metric Density in Finite Hamming Spaces

TL;DR

This paper introduces a new combinatorial rank for subsets in finite Hamming spaces, establishing bounds related to metric properties and defining metrically dense subsets that optimize the representation of distance structures.

Contribution

It defines a novel combinatorial rank based on non-constant columns, providing bounds and characterizations of metric density in finite Hamming spaces, linking algebraic and metric structures.

Findings

Bounds for rank in terms of pairwise distances

Metrically dense subsets minimize rank

Linear subspaces are metrically dense when q is a prime power

Abstract

We introduce a novel concept of rank for subsets of finite metric spaces E^n_q (the set of all n-dimensional vectors over an alphabet of size q) equipped with the Hamming distance, where the rank R(A) of a subset A is defined as the number of non-constant columns in the matrix formed by the vectors of A. This purely combinatorial definition provides a new perspective on the structure of finite metric spaces, distinct from traditional linear-algebraic notions of rank. We establish tight bounds for R(A) in terms of D_A, the sum of Hamming distances between all pairs of elements in A. Specifically, we prove that 2qD_A/((q-1)|A|^2) <= R(A) <= D_A/(|A|-1) when |A|/q >= 1, with a modified lower bound for the case |A|/q < 1. These bounds show that the rank is constrained by the metric properties of the subset. Furthermore, we introduce the concept of metrically dense subsets, which are subsets that minimize rank among all isometric images. This notion captures an extremal property of subsets that represent their distance structure in the most compact way possible. We prove that subsets with uniform column distribution are metrically dense, and as a special case, establish that when q is a prime power, every linear subspace of E^n_q is metrically dense. This reveals a fundamental connection between the algebraic and metric structures of these spaces.