### Further SOMA Update

This website is authored by John Arhin, and reports on my research into SOMAs. I am a post-doctoral student, who has completed his PhD thesis under the supervision of Professor Leonard H. Soicher. Not only is this website an annex to my thesis, but this website also supplements Soicher's webpage SOMA Update, which in turn supplements and updates the paper:
L.H. Soicher, On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares, Electronic J. Combinatorics 6 (1999), #R32, 15 pp.

We shall refer to this paper as [Soi99].

For the benefit of the reader, we give some basic definitions regarding SOMAs here. Note here that these definitions are the same as those used by Soicher in [Soi99].

The work in this website was supported by a grant from the EPSRC NetCA network grant.

## Some thesis results

My PhD thesis has been accepted, and is located here in PDF format. Amongst the results in my PhD thesis are the following:

• Every SOMA(k,n)  (with k ≥ 1) has a unique unrefinable decomposition, which answers Problem 2 of [Soi99] in the negative.
• A polynomial-time algorithm to compute an unrefinable decomposition of an arbitrary SOMA(k,n).
• An indecomposable SOMA(2,n) exists, except when n ≤ 4. This result is based on joint work with M. A. Ollis. In contrast, it is known that there exists two MOLS of order n, or equivalently a Trojan SOMA(2,n), except when n=2 or n=6. So we now know for which n there exists a SOMA(2,n) of any given ud-type.
• We give all the possible ud-types of a SOMA(k,7) here. Soicher, in SOMA Update, has determined all the ud-types of a SOMA(k,n) with n ≤ 6. So we know all the possible ud-types of a SOMA(k,n) with n ≤ 7.
• A generalisation of the Moore-MacNeish Construction of MOLS for SOMAs. See here for further details.

### More results

As of 19th March 2007: the results mentioned above on the unique unrefinable decomposition of a SOMA(k,n) have been generalised in the following paper:

John Arhin, On the structure of 1-designs with at most two block intersection numbers, Designs, Codes and Cryptography, v. 43 n. 2-3, p. 103-114, June 2007.
doi: 10.1007/s10623-007-9067-4.
PDF Preprint.

Updated May 23rd 2009: Recall BCC problem 16.19, which can be found in the British Combinatorial Problem List given here. We have answered this problem in the positive in the following paper:

John Arhin, Every SOMA(n-2,n) is Trojan, Discrete Mathematics (2008),
doi: 10.1016/j.disc.2008.09.050.
PDF Preprint.

This page is maintained by John Arhin.

Last updated: May 23rd 2009.