How to Use
Enter v, k, λ:
- b, r: Derived parameters
- Check: Integer? Fisher?
- Complement: Dual design
Complement Design
Replace each block B with its complement V\B. Get BIBD(v,v-k,λ') where λ'=b-2r+λ. Example: complement of (7,3,1) is (7,4,2). Both are valid BIBDs! This duality provides two designs for the price of one.
Resolvability
A BIBD is resolvable if blocks can be partitioned into parallel classes (each class covers all v points). Necessary: k|v. Resolvable BIBDs give round-robin tournament schedules. Kirkman's schoolgirl problem asks for resolvable (15,3,1).
Step-by-Step Instructions
- 1Enter v, k, λ.
- 2Check b, r integrality.
- 3Verify Fisher.
- 4Compute complement.
- 5Check resolvability.
BIBD Calculator — Frequently Asked Questions
When does a BIBD exist?+
Necessary: b,r integers and b≥v. For k=3,λ=1: iff v≡1,3(mod6). For k=4,λ=1: iff v≡1,4(mod12). In general, necessary conditions aren't sufficient — existence is decided case-by-case for small v and by construction for infinite families.
What's the difference from a general block design?+
BIBD specifically requires: (1) incomplete (k<v, not every point in every block), (2) balanced (every pair in exactly λ blocks). This is a specific, well-studied subclass. Complete block designs (k=v) are trivial.
How are BIBDs used in experiments?+
When you can't test all treatments together (cost, time), use BIBD to ensure every pair of treatments is compared in λ blocks. This gives balanced, unbiased comparisons. R.A. Fisher introduced this for agricultural experiments in the 1920s.
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