Clement W. H. Lam

More coverings by rook domains

Abstract The set Vkn of all n-tuples (x1, x2,…, xn) with xi ϵ, Z k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ⊆ Vkn such that for each x in Vkn, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ⩽ (p − t + 1)pn−r, where p is a prime and n = 1 + t(p r−1 − 1) (p − 1) for some integers t and r. The same method also gives σ(7, 3) ⩽ 216. Another construction gives the inequality σ(n, kt) ⩽ σ(n, k)tn−1 which implies that σ(q + 1, qt) = qq−1tq when q is a prime power. By proving another inequality σ(np + 1, p) ⩾ σ(n, p)pn(p−1), σ(10, 3) ⩽ 5 · 36 and σ(16, 5) ⩽ 13 · 512 are obtained.

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.

A study of two approaches for reconfiguring fault-tolerant systolic arrays

Presents a critical study of two approaches, the classical RC-cut approach and H.T. Kung and M.S. Lams (Proc. 1984 MIT Conf. Advanced Res. VLSI p.74-83, 1984) RCS-cut approach, for reconfiguring faulty systolic arrays. The amount of cell (processing element) redundancy needed to ensure successful reconfiguration into an n*n array is considered. It is shown that no polynomial bounded redundancy is sufficient for the classical approach, whereas O(n/sup 2/log n) redundancy is sufficient for the Kung and Lams approach. The number of faulty cells that can be tolerated in a given array regardless of their locations is characterized and derived. It is shown that, for both approaches, in almost all cases a square array has better fault tolerance than a rectangular array having the same number of cells. A minimal fault pattern in a 2n*2n array with 3n+1 faults that is not reconfigurable into an n*n array using either of the two approaches is established. >

Classification of affine resolvable 2-(27, 9, 4) designs

Abstract All affine resolvable designs with parameters of the design of the hyperplanes in ternary affine 3-space are enumerated. This enumeration implies the classification (up to equivalence), of all optimal equidistant ternary codes of length 13 and distance 9, as well as all complete orthogonal arrays of strength 2 with 3 symbols, 13 constraints and index 3. Up to isomorphism, there are exactly 68 such designs. The automorphism groups and the rank of the incidence matrices over GF(3) are computed. There are six designs with point-transitive automorphism groups, and one design with trivial group. The affine geometry design is the unique design with lowest 3-rank, and the only design with 2-transitive automorphism group.

Resolvable group divisible designs with block size four and group size six

In this paper, we continue the investigation for the existence of resolvable group divisible designs with block size four, group-type h^n and index unity. The necessary conditions for such a design are n>=4, hn=0(mod4) and h(n-1)=0(mod3). The existence of these designs depends mainly on the cases h=1, 2, 3, 6 and 12. Up to now, more than half of these cases have been solved or almost solved except for h=2 and 6. We shall show that the above necessary conditions are also sufficient for h=6 except n=4 and possibly excepting n@?{6,52,54,58,62,68,74,102,114,124}.

Resolvable maximum packings with quadruples

Let V be a finite set of v elements. A packing of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at most one member of F. For fixed v and k, the packing problem is to determine the number of blocks in any maximum packing. A maximum packing is resolvable if we can partition the blocks into classes (called parallel classes) such that every element is contained in precisely one block of each class. A resolvable maximum packing of the pairs of V by k-subsets is denoted by RP(v,k). It is well known that an RP(v,4) is equivalent to a resolvable group divisible design (RGDD) with block 4 and group size h, where h=1,2 or 3. The existence of 4-RGDDs with group-type hn for h=1 or 3 has been solved except for (h,n)=(3,4) (for which no such design exists) and possibly for (h,n)∈{(3,88),(3,124)}. In this paper, we first complete the case for h=3 by direct constructions. Then, we start the investigation for the existence of 4-RGDDs of type 2n. We shall show that the necessary conditions for the existence of a 4-RGDD of type 2n, namely, n ≥ 4 and n ≡ 4 (mod 6) are also sufficient with 2 definite exceptions (n=4,10) and 18 possible exceptions with n=346 being the largest. As a consequence, we have proved that there exists an RP(v,4) for v≡ 0 (mod 4) with 3 exceptions (v=8,12 or 20) and 18 possible exceptions.

A General backtrack algorithm for the isomorphism problem of combinatorial objects

Our aim is to present a practical algorithm for the isomorphism problem that can be easily adapted to any class of combinatorial objects. We investigate the underlying principles of backtrack algorithms that determine a canonical representative of a combinatorial object. We identify the parts of the algorithm that are dependent on the class of combinatorial objects and those parts that are independent of the class. An interface between the two parts is developed to provide a general backtrack algorithm for the isomorphism problem of combinatorial objects that incorporates the technique of branch-and-bound, and that also uses the automorphisms of the combinatorial object to prune the search tree. Our general algorithm incorporates from computational group theory an algorithm known as the base change algorithm. The base change algorithm allows one to recover as much information as possible about the automorphism group when a new branch of the search tree is processed. Thus, it can lead to greater pruning of the search tree. This work is intended to lead to a better understanding of the practical isomorphism algorithms. It is not intended as a contribution to the theoretical study of the complexity of the isomorphism problem.

Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4

abstractThe symmetric class-regular (4,4)-nets having a group of bitranslations G of order four are enumerated up to isomorphism. There are 226 nets with