In this chapter we describe functions to calculate certain matrices
associated with a block design, and the function `BlockDesignEfficiency`

which determines certain statistical efficiency measures of a 1-design.

`PointBlockIncidenceMatrix( `

` )`

This function returns the point-block incidence matrix `N` of the
block design `D`. This matrix has rows indexed by the points of `D`
and columns by the blocks of `D`, with the `(i,j)`-entry of `N` being
the number of times point `i` occurs in `D``.blocks[`

`j``]`

.

The returned matrix `N` is immutable.

gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));; gap> BlockDesignBlocks(D); [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ], [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ], [ 4, 7, 10, 11 ] ] gap> PointBlockIncidenceMatrix(D); [ [ 1, 1, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 1, 1 ], [ 0, 1, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 1, 1, 0 ], [ 0, 1, 0, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 1, 1, 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 1, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 1, 0, 0, 1 ], [ 0, 0, 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 1, 0, 1, 0 ] ]

`ConcurrenceMatrix( `

` )`

This function returns the concurrence matrix `L` of the block design `D`.
This matrix is equal to `NN ^{T}`, where

The returned matrix `L` is immutable.

gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));; gap> BlockDesignBlocks(D); [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ], [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ], [ 4, 7, 10, 11 ] ] gap> ConcurrenceMatrix(D); [ [ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 ], [ 1, 3, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 ], [ 1, 1, 3, 1, 1, 1, 0, 0, 1, 1, 1, 1 ], [ 1, 1, 1, 3, 0, 1, 1, 1, 0, 1, 1, 1 ], [ 1, 1, 1, 0, 3, 1, 1, 1, 0, 1, 1, 1 ], [ 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 1, 1 ], [ 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1 ], [ 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1 ], [ 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 1, 1 ], [ 1, 0, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1 ], [ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 0 ], [ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3 ] ]

`InformationMatrix( `

` )`

This function returns the information matrix `C` of the block design `D`.

This matrix is defined as follows. Suppose `D` has `v` points and `b`
blocks, let `R` be the `vtimesv` diagonal matrix whose `(i,i)`-entry
is the replication number of the point `i`, let `N` be the point-block
incidence matrix of `D` (see PointBlockIncidenceMatrix), and let `K`
be the `btimesb` diagonal matrix whose `(j,j)`-entry is the length of
`D``.blocks[`

`j``]`

. Then the **information matrix** of `D` is
`C:=R-NK ^{-1}N^{T}`. If

The returned matrix `C` is immutable.

gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));; gap> BlockDesignBlocks(D); [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ], [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ], [ 4, 7, 10, 11 ] ] gap> InformationMatrix(D); [ [ 9/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 0 ], [ -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ], [ -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, -1/4 ], [ -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ], [ -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4 ], [ -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4, 0, -1/4, -1/4 ], [ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, 0, -1/4, -1/4, -1/4, -1/4 ], [ -1/4, -1/4, 0, -1/4, -1/4, -1/4, 0, 9/4, -1/4, -1/4, -1/4, -1/4 ], [ -1/4, -1/4, -1/4, 0, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4, -1/4 ], [ -1/4, 0, -1/4, -1/4, -1/4, 0, -1/4, -1/4, -1/4, 9/4, -1/4, -1/4 ], [ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 9/4, 0 ], [ 0, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, -1/4, 0, 9/4 ] ]

`BlockDesignEfficiency( `

` )`

`BlockDesignEfficiency( `

`, `

` )`

`BlockDesignEfficiency( `

`, `

`, `

` )`

Let `D` be a 1-`(v,k,r)` design with `v>1`, let `eps` be a positive
rational number (default: `10 ^{-6}`), and let

`false`

). Then this function returns a record
The component `eff``.A`

contains the A-efficiency measure for `D`,
`eff``.Dpowered`

contains the D-efficiency measure of `D` raised to the
power `v-1`, and `eff``.Einterval`

is a list `[a,b]` of non-negative
rational numbers such that if `x` is the E-efficiency measure of `D`
then `alexleb`, `b-ale``eps`, and if `x` is rational then `a=x=b`.
Moreover `eff``.CEFpolynomial`

contains the monic polynomial over the
rationals whose zeros (counting multiplicities) are the canonical
efficiency factors of the design `D`. If `includeMV``=true`

then
additional work is done to compute the MV- (also called E`'`-) efficiency
measure, and then `eff``.MV`

contains the value of this measure. (This
component may be set even if `includeMV``=false`

, as a byproduct of
other computation.)

We now define the canonical efficiency factors and the A-, D-, E-, and MV-efficiency measures of a 1-design.

Let `D` be a 1-`(v,k,r)` design with `vge2`, let `C` be the information
matrix of `D` (see InformationMatrix), and let `F:=r ^{-1}C`.
The eigenvalues of

If `D` is not connected, then the A-, D-, E-, and MV-efficiency measures
of `D` are all defined to be zero. Otherwise, the **A-efficiency
measure** is `(v-1)/sum _{i=1}^{v-1}1/delta_{i}` (the harmonic mean
of the canonical efficiency factors), the

If `D` is connected, and the MV-efficiency measure is required,
then it is computed as follows. Let `F:=r ^{-1}C` be as before,
and let

gap> D:=DualBlockDesign(AGPointFlatBlockDesign(2,3,1));; gap> BlockDesignBlocks(D); [ [ 1, 2, 3, 4 ], [ 1, 5, 6, 7 ], [ 1, 8, 9, 10 ], [ 2, 5, 8, 11 ], [ 2, 7, 9, 12 ], [ 3, 5, 10, 12 ], [ 3, 6, 9, 11 ], [ 4, 6, 8, 12 ], [ 4, 7, 10, 11 ] ] gap> BlockDesignEfficiency(D); rec( A := 33/41, CEFpolynomial := x_1^11-9*x_1^10+147/4*x_1^9-719/8*x_1^8+18723/128*x_1^7-106\ 47/64*x_1^6+138159/1024*x_1^5-159813/2048*x_1^4+2067201/65536*x_1^3-556227/655\ 36*x_1^2+89667/65536*x_1-6561/65536, Dpowered := 6561/65536, Einterval := [ 3/4, 3/4 ] ) gap> BlockDesignEfficiency(D,10^(-4),true); rec( A := 33/41, CEFpolynomial := x_1^11-9*x_1^10+147/4*x_1^9-719/8*x_1^8+18723/128*x_1^7-106\ 47/64*x_1^6+138159/1024*x_1^5-159813/2048*x_1^4+2067201/65536*x_1^3-556227/655\ 36*x_1^2+89667/65536*x_1-6561/65536, Dpowered := 6561/65536, Einterval := [ 3/4, 3/4 ], MV := 3/4 )

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design manual

November 2011