


Design Theory and Coding Theory / Théorie du design et
théorie des codes (Org: John van Rees)
 FRANK BENNETT, Mount Saint Vincent University, Halifax, Nova Scotia B3M 2J6
Existence of (v, 5, w*)PBDs

In this paper, we investigate the existence of pairwise balanced
designs on v points having blocks of size five, with a distinguished
block of size w, briefly (v,{5,w^{*}},1 )PBDs.
The necessary conditions for a (v,{5,w^{*}},1 )PBD with a distinguished block of
size w are that v ³ 4w+1, v º w º 1 mod 4 and either
v º w mod 20 or v+w º 6 mod 20. Previously, w £ 33
has been studied, and the necessary conditions are known to be
sufficient for w=1, 5, 13 and 21, with 8 possible exceptions
when w £ 33. In this article, we eliminate 3 of these possible
exceptions, showing sufficiency for w=25 and 33. Our main objective
is the study of 37 £ w £ 97, where we establish sufficiency for
w=73, 81, 85 and 93, with 67 possible exceptions with 37 £ w £ 97.
For w º 13 mod 20, we show that the necessary conditions are
sufficient except possibly for w=53, 133, 293 and 453. For
w º 1,5 mod 20, we show the necessary conditions are sufficient
for w ³ 1281,1505, and for w º 9,17 mod 20, we show that
w ³ 2029,2477 is sufficient with one possible exceptional series,
namely v=4w+9 when w º 17 mod 20. We know of no example where
v=4w+9.
In this article, we also study the 4RBIBD embedding problem for
small subdesigns (up to 52 points) and update some results of Bennett
et al. on PBDs containing a 5line.
As an application of our results for w=33 and 97, we establish the
smallest number of blocks in a pair covering design with k=5 when
v º 1 mod 4 with 37 open cases, the largest being for
v=489; hitherto, there were 104 open cases, the largest being
v=2249.
This is joint work with Yanxun Chang, Gennian Ge, and Malcolm
Greig.
 ILIYA BLUSKOV, University of Northern British Columbia
Computing "small" designs

I will discuss some computational aspects of finding small designs in
connection to results on coverings and balanced incomplete block
designs. These include findings from several recent papers, including
joint works with J. Abel and M. Greig on (v,k,l) BIBDs with
k=8,9; joint work with A. Assaf, F. Bennett and M. Greig on the
covering number C(v,5,2) for v=0 mod 4; joint work with R.
Bertolo, H. Hämäläinen and F. Santisi on the covering
number C(v,k,t,m) and some results on the covering number
C_{2}(v,k,t) for k=6, a parameter of interest to lottery players. I
will present some small designs found by either direct constructions or
by assuming some structure and then applying leveldependent
experimental optimization, a method which proved to be successful in
many searches.
 ROBERT CRAIGEN, University of Manitoba, Winnipeg, Manitoba
Power Hadamard matrices

Let n be an integer and f a polynomial. A power Hadamard matrix,
PH(n,f), is an array, H, of integer powers of an indeterminate,
x, with the property that HH^{*} = nI mod f(x). Here H^{*} is the
matrix obtained from H by performing the transpose and then replacing
x with [ 1/(x)].
For example, the following array is a PH(3,1+x+x^{2}):
Power Hadamard matrices are a simple generalization of Butson Hadamard
matrices, which are matrices H whose entries are (complex) roots of
unity and HH^{*}=nI, where H^{*} is the Hermitian adjoint. That is,
BH(n) = PH(n,f), where f is a suitable product of cyclotomic
polynomials. Butson Hadamard matrices, in turn, generalize ordinary
Hadamard matrices, whose entries are ±1 (square roots of unity).
It is also possible to generalize weighing matrices and orthogonal
designs along these lines.
I introduced power Hadamard matrices last summer with a group of
undergraduate and graduate student research assistants. Among other
things, they managed to use these matrices to eliminate the possibility
of projective planes of order 12 having a certain type of internal
structure. We will discuss this and some other interesing connections
between PH's and more classical designtype objects.
 DAVID DRAKE, University of Florida, Gainesville, Florida 32611, USA
Nets with and without ovals or hyperovals

A karc in a (Bruck) rnet is a set of k points such that
each two, but no three are collinear. The existence of a karc
implies that k £ r+1. A karc is called a hyperoval if
k=r+1, an oval if k=r. If there is a 7net of order 10,
its extended binary code would contain exactly one word of weight 8
which is not a hyperoval.
There is a 7net with a hyperoval held by a Desarguesian affine plane
P(D) if and only if the division ring D is of characteristic 2
and cardinality at least 8; P(D) holds a 7net with oval if and
only if D contains a root of x^{3}x^{2}2x + 1.
There is a 7net of order n with a hyperoval, and hence with an
oval, for every n ³ 63. Let n=f_{1}¼f_{t} where each f_{i} is
an odd prime power with f_{i} ³ 7. Then there is a 7net of order
n without a hyperoval. If also f_{i}\not º 0, ±1 mod 7 for
each i, there is a 7net of order n without an oval.
 HADI KHARAGHANI, University of Lethbridge
A review of Ionintype designs

The IoninKharaghani conjecture states that:
For any integers h ¹ 0 and m > 0, if q=(2h1)^{2} is a
prime power, then there exists a symmetric design with parameters

æ è


4h^{2}(q^{m+1}1)
q1

,(2h^{2}h)q^{m},(h^{2}h)q^{m} 
ö ø

. 

This talk is a brief review of the state of this conjecture.
 DON KREHER, Michigan Technical University
Computing transverse tdesigns

Let P = {X_{1},X_{2},¼,X_{r}} be a partition of a set X and let
G be an automorphism group of P. We present an algorithm to
compute the Gorbits of kelement subsets transverse to the
partition P and use it to compute tables of small transverse
Steiner quadruple systems. This is joint work with Kimberly
A. Lauinge.
 CLEMENT LAM, Concordia University
An update on the computer search for a (22,33,12,8,4)BIBD

(joint work with Richard Bilous, John van Rees, and Larry Thiel)
Amongst block designs, the smallest undecided case is the
(22,33,12,8,4)BIBD. If it exists, then its incidence matrix is
contained in a (33,16) doublyeven selforthogonal code (that does
not contain a coordinate of zeros). There are 594 such codes, up to
equivalence. It has been theoretically proven that 116 of these
codes cannot contain the incidence matrix of such a design. For the
remaining 478 codes, an exhaustive clique search may be tried, on the
weight 12 words of a code, to determine whether or not it contains
such an incidence matrix. We give an update on the progress of this
search, and why we may need some volunteers to help finish the search.
 ESTHER LAMKEN, 773 Colby Street, San Francisco, California 94134
Skew SOLSordered room squares

One of the long open questions about Room squares from the 1960's
asks: What is the relationship between orthogonal Latin squares and
Room squares? In this talk, I will describe an answer to this old
question. Let R be a skew Room square in standard position defined
on N È{¥} and let [`(R)] be the skew complement of
R. Let L be the array of pairs formed by the superposition of R
and [`(R)] with the main diagonal deleted and with main diagonal
(i,i) for i Î N. R is said to be SOLSordered if the
pairs in R and [`(R)] can be ordered so that L is the array of
pairs formed by the superposition of a self orthogonal Latin square and
its transpose. We show that with a very small number of possible
exceptions there exist skew Room squares which can be SOLSordered.
 BEN LI, University of Manitoba, Winnipeg, Manitoba R3T 2N2
A parallel algorithm for enumerating covering designs

In this talk, I will describe an algorithm for enumerating covering
designs. This basic ideas of this algorithm were first discussed by
Francois Margot which incorporate linear programming and isomorphism
rejection techniques. I have made several improvements to this
algorithm by introducing additional pruning techniques and by
parallelizing the algorithm. This algorithm improved the lower bounds
for several covering numbers.
 GARY MULLEN, Department of Mathematics, The Pennsylvania State University,
University Park, Pennsylvania 16802 USA
Sets of partially orthogonal latin squares and projective planes

This is joint work with A.D. Keedwell (University of Surrey).
We investigate the construction of sets of latin squares of a given
nonprime power order q which are as close as possible to being
mutually orthogonal sets. Using uniform cyclic neofields, for any even
q > 2 we are able to construct q1 latin squares of order q with
the property that, when we juxtapose any two of them, we obtain 5q4
distinct ordered pairs. More precisely, we obtain 4q3 ordered pairs
that occur exactly once, q1 pairs that occur exactly q3 times,
and q^{2}5q+4 pairs that do not occur.
Thus for example in the case when q=6 and there is not even a pair of
orthogonal latin squares of order 6, we are able to construct 5
latin squares of order 6 so that when any two of them are juxtaposed,
we obtain 26 distinct ordered pairs. And when q=10, using a
uniform cyclic neofield we are able to construct 9 latin squares of
order 10 with the property that when any two of them are juxtaposed
we obtain 46 distinct ordered pairs.
It is conjectured, and very likely true, that Dneofields exist for
any finite order q except for q=2,6. In cases where a Dneofield
exists, one can do even better. For example when q=10, we can
construct 9 latin squares of order 10 so that of the total possible
\binom92 10^{2} = 3600 possible ordered pairs in a complete set of
9 MOLS of order 10, we obtain 2952 distinct ordered pairs. This
represents 82% of the total possible pairs in a complete set.
 RON MULLIN, University of Waterloo, Waterloo, Ontario N2L 3G1
The factorization of certain polynomials in finite fields

The factorization of binomial polynomials over finite fields has
application in the theory of errorcorrecting codes and elsewhere. In
this talk we give explicit factorizations of some of these and other
polynomials and discuss their connection to some conjectures.
 KEVIN PHELPS, Auburn

 VERA PLESS, Mathematics Department, University of Illinois at Chicago,
Chicago, Illinois 606077045, USA
Explicit constructions of families of LDPC codes with girth at
least six

(joint work with JonLark Kim, Irina Perepelitsa and Uri Peled)
There are new families of codes which can be decoded by message passing
softdecision decoding. These codes perform very well. This
performance seems to depend on characteristics of a graph associated to
the code.
LDPC (low density parity check) codes are serious contenders to Turbo
codes in terms of decoding performance. One of the main problems in
LDPC codes is an explicit construction of such codes whose Tanner
graphs have known girth. For a prime power q and an integer m
larger than 1, Lazebnik and Ustimenko give an explicit construction
of a qregular bipartite graph D(m,q) on 2q^{m} vertices, which has
girth at least m+5 for odd m. We regard these graphs as Tanner
graphs of binary codes LU(m,q). We can determine all the parameters
of LU(2,q). We know that their girth is 6 and their diameter is
4. We know that LU(3,q) has girth 8 and diameter 6 and we
conjecture its dimension. We find some interesting LDPC codes by our
partial row construction.
 ROLF REES, Department of Mathematics and Statistics, Memorial University
of Newfoundland, St. John's, Newfoundland A1C 5S7
An application of Skolem sequences to the construction
of pairwise balanced designs

We consider the construction of (v,{3,k})PBDs where v and k
are even and where there is at least one block of size 3. Such
designs seem to be difficult to construct when v is "small"
compared to k (i.e. k^{3/2} £ v £ k^{2}). We describe a
technique by which Skolem and Skolemtype sequences can be used in an
innovative way to construct certain classes of these designs.
 JOE YUCAS, Southern Illinois University, Carbondale, Illinois 62901,
USA
Paley triple arrays

For odd prime powers, Paleytype constructions of triple arrays are
given providing the first infinite family of such arrays.

