A note on the construction of Latin square type designs

Abstract

A PBIB design based on L₂ association scheme with parameters: $\mathrm{v}={\mathrm{p}}^2,\mathrm{b}=\mathrm{p}\left(\mathrm{p}-1\right),\mathrm{r}=\mathrm{p}-1,\mathrm{k}=\mathrm{p},{\mathrm{\lambda }}_1=0,{\mathrm{\lambda }}_2=1,{\text{n}}_1=2\left(\mathrm{p}-1\right),$ ${\text{n}}_2={\left(\mathrm{p}-1\right)}^2$ has been constructed in literature using mutually orthogonal Latin squares (MOLS). In this note we are presenting an alternative method of construction of such designs using the circulant matrix. A method of construction of a PBIB design based on the L₂ association scheme with generalized parameters $\mathrm{v}={\mathrm{p}}^2,\mathrm{ }\mathrm{b}=\mathrm{p}\left\{2s+t\left(p-1\right)\right\},\mathrm{ }\mathrm{r}=2\mathrm{s}+\mathrm{t}\left(\mathrm{p}-1\right),\mathrm{ }\mathrm{k}=\mathrm{p},{\text{ λ}}_1=\mathrm{s},{\text{ λ}}_2=\mathrm{t},{\text{ n}}_1=2\left(\mathrm{p}-1\right){,\mathrm{ }\mathrm{n}}_2={(\mathrm{p}-1)}^2$ where p is a prime is also given.

Keywords:

• Association scheme
• Latin square type designs
• circulant matrix
• discrete Fourier transform matrix

Acknowledgements

The authors are thankful to an anonymous referee and Professor Kishore Sinha for their valuable suggestions in improving the paper.