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Algebraic and Combinatorial Approaches to Designs and Codes - Part II
Org: Thaís Bardini Idalino (Universidade Federal de Santa Catarina, Brazil), Jonathan Jedwab (Simon Fraser University) et Shuxing Li (Simon Fraser University)
[PDF]
Org: Thaís Bardini Idalino (Universidade Federal de Santa Catarina, Brazil), Jonathan Jedwab (Simon Fraser University) et Shuxing Li (Simon Fraser University)
[PDF]
- JIM DAVIS, University of Richmond, VA
Designs with the Symmetric Difference Property [PDF]
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The recently completed search for existence of difference sets in small 2-groups has provided a wealth of data to explore other questions. One classical question asks which designs have the symmetric difference property (the symmetric difference of any three blocks is either a block or the complement of a block). We show in this talk that the groups $C_8 \times C_4^t \times C_2$ all have difference sets whose designs have the symmetric difference property, $t \geq 1$, and that these designs are nonisomorphic to the symplectic designs. Joint with Smith, Hoo, Kissane, Liu, Reedy, Sharma, and Sun.
- HADI KHARAGHANI, University of Lethbridge
A class of balanced weighing matrices and the corresponding association scheme [PDF]
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Balanced weighing matrices with parameters $$
\left(1+18\cdot\frac{9^{m+1}-1}{8},9^{m+1},4\cdot 9^m\right),
$$
for each nonzero integer $m$ is constructed. This seems to be the first infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association scheme.
This is joint work with Thomas Pender and Sho Suda.
- ZEYING WANG, Michigan Technological University, MI
New necessary conditions on (negative) Latin square type partial difference sets in abelian groups [PDF]
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A partial difference sets (in short, PDS) with parameters ($n^2$, $r(n-\epsilon)$, $\epsilon n+r^2-3\epsilon r$, $r^2-\epsilon r$) is called a {\it Latin square type} PDS if $\epsilon=1$ (respectively, a {\it negative Latin square type} PDS if $\epsilon=-1$). Recently we obtained some restrictions on the parameter $r$ of a (negative) Latin square type partial difference set in an abelian group of order $a^2 b^2$, where $\gcd(a,b)=1$, $a>1$, and $b$ is an odd positive integer $\ge 3$. As far as we know no previous general restrictions on $r$ were known. Our restrictions are particularly useful when $a$ is much larger than $b$.
- IAN WANLESS, Monash University, Australia
Omniversal Latin squares [PDF]
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A partial transversal of a Latin square is a set of entries in which
no row, column or symbol is repeated. It is maximal if it is not
contained in a larger partial transversal. A Latin square of order $n$
is omniversal if it possesses a maximal partial transversal of every
size from $\lceil\frac{n}{2}\rceil$ to $n$. We show that omniversal
Latin squares exist iff $n\not\equiv2\bmod4$ and $n\notin\{3,4\}$. We
also show that group tables are very far from omniversal (as are
random Latin squares). In the process we encounter an interesting
problem in combinatorial group theory.
- XIANDE ZHANG, University of Science and Technology of China
Optimal ternary constant weight codes in $l_1$-metric [PDF]
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In this talk, we discuss our recent progress on the existence of optimal ternary constant weight codes in $l_1$-metric. We determine the maximum size of ternary codes of constant weight $w$ and distance $2w-2$ for all large length $n$. For distance $2w-4$, we determine the coefficients of $n^2$ by constructing asymptotially optimal codes. The motivation of studying constant weight codes in $l_1$-metric is from data storage in live DNA.