Switchingdistinct

Switchingdistinct is a term used to describe a property of a set of binary configurations under a family of single-bit toggle operations. In systems modeled by binary switches, a configuration is a vector in {0,1}^n, describing which switches are on or off.

Formal definition: A subset S ⊆ {0,1}^n is switchingdistinct if the Hamming-1 neighborhoods around its elements are pairwise disjoint. The Hamming-1 neighborhood of a configuration x is the set of all configurations obtainable by flipping exactly one coordinate of x. Equivalently, for any x ≠ y in S, the Hamming distance d(x,y) between x and y is at least 3, since distance 1 or 2 would imply overlapping one-step toggles between the two configurations.

Examples: In n = 3, the set S = {000, 111} is switchingdistinct, because the radius-1 neighborhoods {000,001,010,100} and {111,110,101,011} are disjoint. More generally, any binary code with minimum distance at least 3 yields a switchingdistinct set.

Applications: The concept is relevant in error detection and fault-tolerant switch networks, where it is important to distinguish configurations with minimal switching. It also ties to coding theory, particularly to ideas about sphere packing and distinct representations under single-bit toggling.

Properties: Switchingdistinct implies a minimum Hamming distance of at least 3 among elements of the set. The maximal size of a switchingdistinct set in length n relates to the size of binary codes with distance 3 (A(n,3)).

See also: error-correcting codes, Hamming distance, binary codes, sphere packing.