# Which Latin square is the loneliest?

Presented by:

Nick Cavenagh, the University of Waikato
Date:

Tue 22 Sep, 4:00 pm - 5:00 pm
Venue:

Building 67, Room 442 Given two Latin squares $L_1$ and $L_2$ of the same order, the {\em hamming distance} between $L_1$ and $L_2$ gives the number of corresponding cells containing distinct symbols. If we think of Latin squares as sets of ordered triples, this is given by $|L_1\setminus L_2|$. Given a specific Latin square $L_1$, we may wish to know a Latin square $L_2$ which is closest to it; i.e. for which the Hamming distance is minimized.

Equivalently, we may ask for the size of the smallest Latin trade within a given

__latin__square.It is known that the back circulant Latin square of order $n$ (the operation table for the integers modulo $n$) has Hamming distance at least $e\log{n}+3$ to any other Latin square. We explore whether the back circulant Latin square is the loneliest of all Latin squares; i.e.has greatest minimum Hamming distance to any other Latin square. Joint work with R.

__Ramadurai.__