minimumdistance

Minimum distance is a general term used across mathematics, computer science, and engineering to denote the smallest separation between two objects of interest. Although the precise meaning depends on context, the idea is to quantify how close two points, sets, codewords, or signals can be.

In geometry and analysis, the distance between two points p and q in Euclidean space is the norm of their difference, usually d(p, q) = ||p − q||. The minimum distance between two sets A and B is defined as d(A, B) = inf{ ||a − b|| : a in A, b in B }. If A and B are closed and disjoint in a Euclidean space, this infimum is attained and positive. The closest-distance problem between a finite set of points seeks the smallest pairwise distance among all points.

In coding theory, the minimum distance of a code C, denoted d_min(C), is the smallest Hamming distance between any pair of distinct codewords. It determines error-detecting and error-correcting capabilities: a code with d_min can detect up to d_min − 1 errors and, in general, correct up to floor((d_min − 1)/2) errors. For linear codes, d_min equals the minimum weight among all nonzero codewords.

In digital communications and signal processing, the relevant notion is often the minimum Euclidean distance between signal points (constellation points). This distance governs the error probability under noise and affects the design of modulation schemes.

Computationally, problems involving minimum distance include finding the closest pair of points in a set, solvable by efficient algorithms with near-linear time in many cases. Minimum distance is also used in safety contexts to specify the required separation between vehicles or from hazards.

Overall, minimum distance provides a concise measure of proximity that influences performance, reliability, and safety across disciplines.