Block Design Calculator

Fisher's inequality: b ≥ v

CalculatorsFreeNo Signup
4.3(508 reviews)
All Tools

Loading tool...

About Block Design Calculator

A block design calculator computing parameters (v,b,r,k,λ) where v points are arranged in b blocks of size k, each point in r blocks, every pair in λ blocks. Fisher's inequality: b≥v. Verifies necessary conditions. Client-side.

Block Design Calculator Features

  • Parameter check
  • Fisher's inequality
  • Replication
  • Pair balance
  • Symmetric
Block design (v,b,r,k,λ): v points, b blocks of size k, each point in r blocks, each pair in λ blocks. Relations: bk=vr and λ(v-1)=r(k-1). Fisher's inequality: b≥v. Symmetric design: b=v (implies r=k).

How to Use

Enter v, k, λ:

  • Parameters: Compute b, r
  • Fisher: Check b≥v
  • Symmetric: Check b=v

Necessary Conditions

For a (v,b,r,k,λ)-BIBD: (1) bk=vr (counting). (2) λ(v-1)=r(k-1) (pair counting). (3) b≥v (Fisher). (4) r≥k. These are necessary but not always sufficient for existence.

Famous Designs

(7,7,3,3,1)=Fano plane (symmetric). (13,26,6,3,1)=Steiner triple. (16,20,5,4,1)=affine plane complement. (6,10,5,3,2)=unique. The Handbook of Combinatorial Designs catalogs thousands.

Step-by-Step Instructions

  1. 1Enter v, k, λ.
  2. 2Compute b, r.
  3. 3Check Fisher.
  4. 4Verify conditions.
  5. 5Identify type.

Block Design Calculator — Frequently Asked Questions

What are the fundamental relations?+

bk=vr (total incidences counted two ways) and λ(v-1)=r(k-1) (pairs through one point). Given any three of {v,b,r,k,λ}, the other two are determined. Usually we specify (v,k,λ) and derive (b,r).

What's Fisher's inequality?+

For any BIBD with k<v: b≥v (number of blocks ≥ number of points). Equality gives a symmetric design. Proof uses the rank of the incidence matrix. This is one of the most fundamental results in combinatorics.

Where are block designs used?+

Agricultural experiments (balancing treatments across plots), clinical trials, communication networks (code design), tournament scheduling, and statistical sampling. They ensure every pair of treatments is compared equally often.

More Calculators Tools

Explore all 639 other calculators tools available on Generatr

1-Sample Z-Test Calculator2-Sample Z-Test Calculator3D Surface Area CalculatorA/B Testing Significance CalculatorAbsolute Value CalculatorAbundant Number CheckerAccelerated Aging CalculatorAchromatic Number CalculatorAcyclic Chromatic Number CalculatorAdditive Persistence CalculatorAge CalculatorAging Test CalculatorAlbertson Index CalculatorAlgebraic Connectivity CalculatorAliquot Sequence CalculatorAliquot Sum CalculatorAlternating Series Test CalculatorAmicable Number FinderAnnuity CalculatorAPY CalculatorAquarium CO2 Drop CheckerArea CalculatorArea Under Curve CalculatorArithmetic Sequence CalculatorArmstrong Number CalculatorASCVD Risk CalculatorAspect Ratio CalculatorAtom Bond Connectivity CalculatorAugmented Eccentric Connectivity CalculatorAugmented Zagreb Index Calculator
View all 639 calculators tools
Share this tool: