ym1: Constructs Youden-m square designs and provides parameters of...

In PBIBD: Partially Balanced Incomplete Block Designs

Constructs Youden-m square designs and provides parameters of the corresponding PBIB design

Description

If the same number of rows, say t, are omitted from the top and bottom of a Circulant matrix such that at least two rows remain, the resulting arrangement forms a Youden-m square.

(A) For even-ordered Circulant matrices (order v \geq 4 and even), the columns of the resulting Youden-m square constitute a PBIB design with parameters:

• b = v, r = k = v - 2t

• \lambda_1 = v - 2(t + 1)

• \lambda_{m - i} = v - 2t - 1 - 2i, for i = 0, 1, \ldots, t - 1

• \lambda_t = \lambda_{t+1} = \ldots = \lambda_{m - t} = v - 4t

• If t \geq 3, then \lambda_i = v - 2(t + i) for i = 2, 3, \ldots, t - 1

(B) For odd-ordered Circulant matrices (order v \geq 5 and odd), the columns of the resulting Youden-m square constitute a PBIB design with parameters:

• b = v, r = k = v - 2t

• \lambda_1 = v - 2t - 1

• \lambda_{m - i} = v - 2(t + 1) - i, for i = 0, 1, \ldots, t - 1

• \lambda_{m - (t - 1) - i} = \lambda_{m - (t - 1)} - i, for i = 0, 1, \ldots, t - 1

• \lambda_2 = \lambda_3 = \ldots = \lambda_{m - 2t + 1} = \lambda_{m - 2t + 2}

Usage

ym1(n, t)

Arguments

n	Order of the circulant matrix, which is also the number of treatments.
t	Number of rows to omit from both the top and bottom of the circulant matrix.

Value

The function returns the Youden-m square design and the parameters of the PBIB design formed by taking its incomplete columns as blocks.

Author(s)

Kush Sharma, Davinder Kumar Garg

Examples

ym1(6, 1)

PBIBD documentation built on June 8, 2025, 10:10 a.m.