CanaDAM 2011 University of Victoria, May 31 - June 3, 2011 www.cms.math.ca//2011

Designs and Codes II
Org: Peter Dukes (University of Victoria)
[PDF]

PETER DUKES, University of Victoria
Injection Codes  [PDF]

An $n$-ary {\em injection code} of length $m$ and minimum distance $d$ is a set $\Gamma$ of injections from an $m$-set (of positions) into an $n$-set (of symbols) such that any two different words in $\Gamma$ have Hamming distance $\ge d$. When $n=m$, an injection code becomes a {\em permutation code}. And when $\Gamma$ meets a certain upper bound, it becomes an {\em ordered design}.

This talk will survey my preliminary observations on injection codes.

The Gramian of mutually unbiased Hadamard matrices  [PDF]

Two Hadamard matrices $H$ and $K$ of order $n$ are called {\em unbiased} if the absolute value of all the entries of $HK^t$ equal $\sqrt{n}$. The Gramian of any ordered set of mutually unbiased Hadamard matrices contains very interesting configurations including some 3-class symmetric association schemes. We will concentrate on the case where $n=4^n$ for this talk.

ESTHER LAMKEN, San Francisco
Existence results for Howell cubes  [PDF]

In this talk, I will describe some new existence results for Howell cubes of even order. We construct Howell cubes, $H_3(2n,2n + \alpha)$, for $\alpha =2,4,6$. I will also describe related results for 3 dimensional frames which are used in our constructions. This is joint work with Jeff Dinitz and Greg Warrington.

DAVID PIKE, Memorial University of Newfoundland
Hamilton cycles in restricted block-intersection graphs  [PDF]

Given a BIBD$(v,k,\lambda)$ with block set $\cal B$, its $i$-block-intersection graph is the graph having vertex set $\cal B$ such that two vertices $B_1$ and $B_2$ are adjacent if and only if $|B_1 \cap B_2| = i$. It has been known since 1999 that the 1-block-intersection graph of any $\lambda$-fold triple system on $v \geq 12$ points is Hamiltonian. We now consider restricted block-intersection graphs of BIBDs with larger block sizes.

ALEX ROSA, McMaster University
Circulants as signatures of cyclic Steiner triple systems  [PDF]

With each cyclic Steiner triple system we associate a certain set of circulants called its {\it signatures}. From among the set of all circulants of appropriate degree, a circulant is a signature if it is a signature of some cyclic Steiner triple system; otherwise, it is a nonsignature. We show the existence of nonsignatures, and discuss questions related to the enumeration of signatures and nonsignatures. (Joint work with Mariusz Meszka.)

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