SEMINAR:
Groups and Combinatorics Seminars
Tue, 04 Dec 2012 13:00 - Maths Lecture Room 2
Neil Gillespie and Daniel Hawtin
Groups and Combinatorics Seminar
Neil Gillespie (UWA)
will speak on
Completely regular codes with large minimum distance
and
Daniel Hawtin (UWA) will speak on Elusive Codes in Hamming Graphs
at 1pm Tuesday 4th of December in MLR2
Abstracts:
Completely regular codes with large minimum distance: In 1973 Delsarte
introduced completely regular codes as a generalisation of perfect
codes. Not only are completely regular codes of interest to coding
theorists due to their nice regularity properties, but they also
characterise certain families of distance regular graphs. Although no
complete classification of these codes is known, there have been several
attempts to classify various subfamilies. For example, Borges, Rifa and
Zinoviev classified all binary non-antipodal completely regular codes.
Similarly, in joint work with Praeger, we characterised particular
families of completely regular codes by their length and minimum
distance, and additionally with Giudici, we also classified a family of
completely transitive codes, which are necessarily completely regular.
In this work with Praeger, and also with Giudici, the classification
given by Borges, Rifa and Zinoviev was critical to the final result.
However, recently Rifa and Zinoviev constructed an infinite family of
non-antipodal completely regular codes that does not appear in their
classification. This, in particular, led to a degree of uncertainty
about the results with Praeger and with Giudici. In this talk I
demonstrate how I overcame this uncertainty by classifying all binary
completely regular codes of length m and minimum distance $ elta$ such
that $ elta>m/2$.
Elusive Codes in Hamming Graphs:
We consider a code to be a subset of the vertex set of a Hamming graph.
We examine elusive pairs, code-group pairs where the code is not
determined by knowledge of its set of neighbours. We provide an infinite
family of elusive pairs, where the group in question acts transitively
on the set of neighbours of the code. In our examples, we find that the
alphabet size always divides the length of the code, and prove that
there is no elusive pair for the smallest set of parameters for which
this is not the case.
For more information:
Michael Giudici
giudici@maths.uwa.edu.au
Starts : Tue, 04 Dec 2012 13:00
Ends : Tue, 04 Dec 2012 13:45
Last Updated : Mon, 03 Dec 2012 07:42