E-best discrepancy matrices

This page gives all E-best discrepancy matrices (up to isomorphism) of order v<=15 for each possible row sum (i.e. concurrence excess) q>0. A discrepancy matrix for an incomplete block design (block size k<v=number of treatments) describes departure of treatment concurrences from complete symmetry in that design. E-best discrepancy matrices describe departures that assure an E-optimal design within the binary, equireplicate class. For all matrices here, E-optimality in fact holds over the entire class including nonbinary and unequally replicated designs, provided the block size k is greater than 2. Details may be found in the paper Optimal Incomplete Block Designs.

For k=2, the general result for E-optimality in the binary, equireplicate class can currently be strengthened in only some cases. Interestingly, for k=2, discrepancy matrices directly describe not just E-optimal information matrices, but the designs themselves. Given b (number of blocks), v, and k=2 such that bk is a multiple of v, let L be the integer part of bk(k-1)/v(v-1)=2b/v(v-1). Then for discrepancy matrix D, the number of blocks containing treatments i and i' is the (i,i') entry in the matrix D+L. This fails only if L=0 and D contains a -1 (all matrices here have smallest element no less than -1). Whenever a list of E-best discrepancy matrices fails to have a member with smallest element 0, the best subject to having all nonegative entries has been determined, and all matrices as E-good or better than that matrix have also been included.

For some other values of v,b,k with k>2 no design exists for any of the corresponding E-best discrepancy matrices. In some cases the best discrepancy matrix for which a design does exist has been determined, and all matrices as good as or better than that matrix have been included.

Each file below is an R-list of discrepancy matrices compressed by bzip2. After downloading and unzipping a file, the list can be read into R by, for example for v=10, q=6, the command load("discmatrices10_6.R"). The loaded list is named discmatrices, and the matrices are discmatrices[[1]], discmatrices[[2]], and so on. The R freeware is available here.

The matrices in each file are ordered by E-value. No other ordering should be assumed. The order is the same as in the xml versions, stored as block designs with k=2, available at designtheory.org

This page maintained with support from the National Science Foundation.

Number of files = 61

Last updated: 2005-08-31