fixedS

FixedS is a term used in the theory of combinatorial designs and sampling to denote a fixed-size subset system with uniform incidence properties. In its core formulation, one starts with a finite universe U of size v and a collection F of k-element subsets of U. The system (U, F) is called a fixedS design when every element of U lies in exactly r blocks of F, and any pair of distinct elements occurs together in a constant number λ of blocks. When the stronger condition holds that every pair occurs in exactly λ blocks, the system is a fixedS balanced incomplete block design (BIBD). The standard parameter relations vr = bk and r(k − 1) = λ(v − 1) appear as necessary conditions for existence.

Properties of fixedS designs include fixed block size, uniform coverage of elements, and often a symmetric or highly regular incidence structure. These features make fixedS designs useful as idealized models for fair sampling, test-set construction, and distributed key distribution schemes in networks. They also connect closely with broader topics in combinatorial design, such as Steiner systems and projective-plane configurations.

Construction of fixedS designs can be achieved via known combinatorial families (Steiner systems, finite projective planes) or by randomized methods that yield approximate fixedS properties. Existence is subject to arithmetic constraints on the parameters (v, k, r, λ), and not all combinations are realizable.

Applications of fixedS designs span data sampling, experimental design, network design, and cryptographic key predistribution, where uniform participation and balanced coverage are desirable. See also combinatorial design, block design, Steiner system, and sampling theory.