This page gives E-optimal block designs with v treatments in b blocks of constant block size k, such that v divides bk, v<=15, and 3<=k<=v/2, excluding cases where BIBDs exist. For $v/2<k<=v-2$, E-optimal designs are the complements of those cataloged here. A few (v,b,k) combinations with k>v/2 are such that there is no connected complement (these arise as complements of single-replicate designs); the E-optimal designs for these settings are also included here. In all cases the E-optimality holds over the entire class of all competitors with the same v,b,k, including nonbinary and unequally replicated designs. The lists of designs are in DTRS external representation protocol version 2.0 format, compressed by bzip2. Each file has content types i and g, which mean that each design in that file has, respectively, <indicators> and <automorphism_group> subtrees. Some files have content type m, which means that all E-optimal designs up to isomorphism of information matrices are listed. A resolution of a design is given whenever one exists.
The parameter q is the concurrence excess for the design setting. Write r=bk/v and let lambda denote the integer part of r(k-1)/(v-1). Then q=r(k-1)-lambda*(v-1). The information matrix for any binary, equireplicate design can be written as the sum of a completely symmetric matrix and a constant times a discrepancy matrix. The discrepancy matrix is a symmetric, integer-valued matrix with zero diagonal and constant row sums q. Details of this relationship, and the correspondence between E-optimal block designs and E-best discrepancy matrices, can be found in the paper Optimal Incomplete Block Designs. Most designs in this catalog have discrepancy matrices which are listed as graphs (i.e. block designs with block size 2) in discrepancy matrix catalog. The id for each such design contains the corresponding graph id. Notes attached to the designs indicate when E-optimality has been established by methods other than use of the discrepancy matrix; these say "Found by enumeration," typically meaning some combination of theory to eliminate many possibilities with an enumeration of remaining competitors, as explained in Optimal Incomplete Block Designs.An E-optimal design d for (v,b,k) is BIBD-extendable if dUd* is E-optimal in (v,b+b*,k) for any b* for which there is a BIBD d* in (v,b*,k). Lists of designs here are BIBD-extendable if ext=1 and otherwise are not. The BIBD-extendable designs are exactly those with E-best discrepancy matrices. Through the concept of BIBD-extendability, this catalog solves the E-optimal design problem for v<=11, 3<=k<=v-2, v divides bk, in the sense of determining all E-optimal designs up to isomorphism of information matrices, save for the case (v,q,k)=(10,6,5) (where the existence of designs for some E-best graphs is still open). E-optimal designs which are not BIBD-extendable are cataloged only when all possibilities for extendable designs have been proven nonexistent.
For 12<=v<=15, the goal has been to solve the E-optimality problem in the sense of determining at least one BIBD-extendable E-optimal design for each (v,q,k) and corresponding smallest possible b, and to find at least one E-optimal design when no extendable design exists. For v=12,13,15 this has been accomplished save for one case each: (v,q,k) = (12,2,5), (13,4,5), (15,12,6). There are several cases open for v=14. Open cases appear in the catalog as files with #designs = 0. These files contain brief notes only. New solutions are welcome and will be added to the catalog!
For k=2, a complete solution to the E-optimality problem for v<=15 within the binary, equireplicate class (up to isomorphism of information matrices) may be found at discrepancy matrix catalog.
This page constructed with support from the U.S. National Science Foundation.
Number of files = 343
| v | q | k | b | ext | content type | # designs | |
|---|---|---|---|---|---|---|---|
| 5 | 2 | 3 | 5 | 1 | i g m | 1 | download |
| 6 | 1 | 3 | 6 | 1 | i g m | 1 | download |
| 6 | 1 | 3 | 16 | 1 | i g m | 2 | download |
| 6 | 1 | 3 | 26 | 1 | i g m | 2 | download |
| 6 | 1 | 3 | 36 | 1 | i g m | 3 | download |
| 6 | 1 | 4 | 3 | 1 | i g m | 1 | download |
| 6 | 1 | 4 | 18 | 1 | i g m | 3 | download |
| 6 | 2 | 3 | 12 | 1 | i g m | 1 | download |
| 6 | 2 | 3 | 22 | 1 | i g m | 2 | download |
| 6 | 2 | 4 | 6 | 1 | i g m | 1 | download |
| 6 | 2 | 4 | 21 | 1 | i g m | 2 | download |
| 6 | 3 | 3 | 8 | 0 | i g m | 3 | download |
| 6 | 3 | 3 | 18 | 1 | i g m | 1 | download |
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| 6 | 3 | 4 | 9 | 1 | i g m | 1 | download |
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| 6 | 4 | 4 | 12 | 1 | i g m | 1 | download |
| 7 | 2 | 5 | 7 | 0 | i g m | 1 | download |
| 7 | 2 | 5 | 28 | 1 | i g m | 1 | download |
| 7 | 4 | 5 | 14 | 1 | i g m | 1 | download |
| 8 | 1 | 3 | 48 | 1 | i g m | 5 | download |
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| 8 | 2 | 4 | 20 | 1 | i g m | 2 | download |
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| 15 | 6 | 13 | 150 | 1 | i g m | 234 | download |
| 15 | 8 | 3 | 20 | 1 | i g | 1 | download |
| 15 | 8 | 4 | 45 | 1 | i g | 2 | download |
| 15 | 8 | 5 | 6 | 0 | i g m | 1 | download |
| 15 | 8 | 5 | 27 | 1 | i g | 1 | download |
| 15 | 8 | 6 | 25 | 1 | i g | 1 | download |
| 15 | 8 | 13 | 60 | 1 | i g m | 2 | download |
| 15 | 8 | 13 | 165 | 1 | i g m | 19 | download |
| v | q | k | b | ext | content type | # designs | |
| 15 | 10 | 3 | 25 | 1 | i g m | 1 | download |
| 15 | 10 | 4 | 30 | 1 | i g m | 1 | download |
| 15 | 10 | 5 | 18 | 0 | i g | 1 | download |
| 15 | 10 | 5 | 39 | 1 | i g m | 1 | download |
| 15 | 10 | 6 | 5 | 0 | i g m | 1 | download |
| 15 | 10 | 6 | 40 | 1 | i g m | 1 | download |
| 15 | 10 | 13 | 75 | 1 | i g m | 1 | download |
| 15 | 12 | 3 | 30 | 1 | i g m | 1 | download |
| 15 | 12 | 4 | 15 | 1 | i g m | 1 | download |
| 15 | 12 | 5 | 9 | 0 | i g m | 42 | download |
| 15 | 12 | 5 | 30 | 1 | i g m | 1 | download |
| 15 | 12 | 5 | 51 | 1 | i g m | 1 | download |
| 15 | 12 | 6 | 20 | 1 | -none- | 0 | download |
| v | q | k | b | ext | content type | # designs | |
| 15 | 12 | 13 | 90 | 1 | i g m | 1 | download |
Last updated: 2006-09-08 14:04:52.694175