Larry H. Thiel

The nonexistence of ovals in a projective plane of order 10

This paper reports the result of a computer search which shows that there is no oval in a projective plane of order 10. It gives a brief description of the search method as well as a brief survey of other possible configurations in a plane of order 10.

The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code

An extremal self-dual doubly-even binary (n,k,d) code has a minimum weight d=4/spl lfloor/n/24/spl rfloor/+4. Of such codes with length divisible by 24, the Golay code is the only (24,12,8) code, the extended quadratic residue code is the only known (48,24,12) code, and there is no known (72,36,16) code. One may partition the search for a (48,24,12) self-dual doubly-even code into three cases. A previous search assuming one of the cases found only the extended quadratic residue code. We examine the remaining two cases. Separate searches assuming each of the remaining cases found no codes and thus the extended quadratic residue code is the only doubly-even self-dual (48,24,12) code.

A computer search for finite projective planes of order 9

Abstract There are four known finite projective planes of order 9. This paper reports the result of a computer search which shows that this list is complete. The computer search starts by generating all 283,657 non-isomorphic latin squares of order 8. Each latin square gives 27 columns of the incidence matrix. Another program attempts to complete each of these incidence matrices to 40 columns. Only 21 of them can be so completed, giving rise to 326 matrices of 40 columns. A third computer program attempts to complete the rest of the matrices. One of the 326 does not complete. The rest complete each to a unique matrix. An isomorphism testing program is then applied to the 325 complete matrices, creating a certificate for each matrix, as well as its collineation group. The certificates are then compared with the known planes and no new ones found. As a further evidence of the correctness of the computer programs, this paper also shows that the computer results are consistent with those expected by using information about the known planes and their associated latin squares.

On the number of 8 x 8 Latin squares

Abstract The numbers of non-isomorphic Latin squares of order 8 under the action of various different symmetry groups have been reported by Wells in 1967, by Brown in 1968 and by Arlazarov and coworkers in 1978. The result of Wells has also been independently verified by Bammel and Rothstein in 1975. As an intermediate step towards finding a complete list of projective planes of order 9, we have to generate all the Latin squares of order 8. We find numbers that agree with Wells, but different from those of Brown and Arlazarov and coworkers. These new numbers are reported in this note. We also give the distributions of these numbers according to the size of the automorphism group, which allows one to easily cross check the results.

There is no (46, 6, 1) block design*

Abstact: In this paper we show that a (46, 6, 1) design does not exist. This result was obtained by a computer search. In the incidence matrix of such a design, there must exist a “c4” configuration—6 rows and 4 columns, in which each pair of columns intersect exactly once, in distinct rows. There can also exist a “c5” configuration with 10 rows and 5 columns, in which each pair of columns intersect exactly once, in distinct rows. Thus the search for (46, 6, 1) designs can be subdivided into two cases, the first of which assumes there is no “c5”, and the second of which assumes there is a “c5”. After completing the searches for both cases, we found no (46, 6, 1) design.

The nonexistence of code words of weight 16 in a projective plane of order 10

Abstract The weight enumerator of the binary error-correcting code generated by the rows of the incidence matrix of a projective plane of order 10 is completely determined once the numbers of code words of weight 12, 15, and 16 are known. The search for such a projective plane starting from code words of weight 16 can be divided into six cases. In his 1974 Ph.D. thesis at Berkeley, Carter finished the search for four of these cases as well as part of the fifth. This note reports the results of a computer search for the remaining cases. No projective plane of order 10 was found.

Three improvements to the BLASTP search of genome databases

The BLASTP program is a search tool for databases of protein sequences that is widely used by biologists as a first step in investigating new genome sequences. BLASTP finds high-scoring local alignments (q/sub i/q/sub i+1/...q/sub i+k//spl par/s/sub j/s/sub j+1/...s/sub j+k/) without gaps between a query sequence q and sequences s in the database. The score of an alignment is the sum of the scores of individual alignments q/sub i+t//spl par/s/sub j+t/ between amino acids that make up the protein. These individual scores come from a scoring matrix modeling the rate of evolutionary mutation. Here we provide a detailed description of the original program and three separate optimisations to it. BLASTP consists of three steps, that we call neighbourhood construction, hit detection, and hit extension. The three optimisations target hit extension since it accounts for 93% of the execution time. The first optimisation alters the data representation of the query sequence and the related code for indexing the scoring matrix. The second optimisation performs extensions in step-sizes of two rather than one. The third optimisation forstalls the calling of the hit extension step in cases that are unlikely to lead to a high-scoring alignment. Individually the three optimisations show speed ups of 15%, 48%, and 63% respectively.

On quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs

New quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs are constructed by embedding known designs into symmetric designs.