## Coding, Cryptology and Combinatorial Designs (15 May - 11 Jun 2011)

Jointly organized with Nanyang Technological University

## ~ Abstracts ~

Construction of entropy optimal real orthogonal and complex inverse orthogonal matrices
K.T. Arasu, Wright State University, USA

The optimization bounds for the Shannon's entropy of a real orthogonal matrix are found for several classes of orthogonal matrices. In the cases when a Hadamard matrix exists, the entropy is shown to achieve its maximum value. For other orders for which the corresponding Hadamard matrix does not exist, sharper bounds on the entropy are obtained. A general algorithm for the optimization in case of other equivalent definitions of entropy is provided based on the prime factorization of the order of the matrix. The algorithm works effectively for other equivalent definitions of entropy of an orthogonal matrix. In case of complex inverse orthogonal matrix, a construction is given which achieves the bound.

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Characterizing graphs by their spectrum
Aart Blokhuis, Eindhoven University of Technology, The Netherlands

The spectrum of the adjacency matrix of a graph reveals a lot of the structure, but still there a relatively few examples of nice graphs that are characterized by their spectrum. In the talk we will illustrate a number of techniques that can be used by showing that there are unique graphs with the following spectra:

A graph on 60 points with spectrum 14 (1), 2 (40), -4 (10) and -6 (9),

This graph was put as a challenge to Andries Brouwer at the occasion of his 60th birthday by Willem Haemers,

A graph on 55 points with spectrum 4 (1), -2 (10), 1\pm\sqrt3 (10) and (3\pm \sqrt5)/2 (12).

This was put as a challenge to me by Andries Brouwer, the corresponding graph has PSL(2,11) acting on it, and has no induced triangles, fourcycles, sixcycles, or sevencycles.

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On counting subsets over finite fields
Jiyou Li, Shanghai Jiao Tong University, China

In this talk, we will discuss the problem of counting certain subsets over finite fields. A new combinatorial approach to this problem will be presented. As an illustration of arithmetical applications, several examples will also be given.
This is a joint work with Prof. Wan Daqing.

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Asymptotic existence of combinatorial designs
Alan Ling, University of Vermont, USA

Over the last 40 years, various works have been done on the asymptotic results in combinatorial design theory. In particular, balanced incomplete block designs, pairwise balanced designs, group divisible designs, resolvable balanced incomplete block designs, edge-colored graph designs, resolvable graph designs and various other problems have been considered by various authors. In this talk, we give a survey of the recent results and also discuss new results, and open problems in this challenging area.

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Two results on planar functions
Alexander Pott, Otto-von-Guericke-Universität Magdeburg, Germany

Motivated by applications in cryptography, the investigation of almost perfect nonlinear functions F on finite fields of even characteristic attracted a lot of attention: A function F is called almost perfect nonlinear if F(x+a)-F(x)=b has at most two solutions for all nonzero a and all b. In geometry, functions F in odd characteristic such that F(x+a)-F(x)=b has always precisely one solution have been investigated under the name of planar (or perfect nonlinear) functions. Most planar functions correspond to commutative semifields, and all planar functions give rise to relative difference sets in elementary-abelian groups.

There are also semifields of even order, which correspond to relative difference sets in groups of exponent 4, but these do not correspond to planar functions.

In my talk, I will briefly discuss these three concepts:

- almost perfect nonlinear functions
- perfect nonlinear functions (related to elementary-abelian relative difference sets and commutative semifields)
- semifields of even order (related to relative difference sets in groups of exponent 4).

The author believes that some methods which have been applied so far to just one of these problems may be also useful for the others.

Following these more general remarks, I will present two theorems on planar functions: One of these is related to "sub"-planar functions, showing an interesting connection between two families of commutative semifields. The second theorem relates a problem on planar functions to a problem about curves.

Authors which also contributed to these results are Yue Zhou (Ph.D. student in Magdeburg), Gohar Kyureghyan (Magdeburg) and Ferruh Ozbudak (Ankara).

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Construction of combinatorial structures using rational idempotents
Ken Smith, Sam Houston State University, USA

We use the concept of rational idempotents in a group ring to construct difference sets, partial difference sets, relative difference sets and circulant weighing matrices.

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Permutation polynomials and commutative semifields
Guobiao Weng, Dalian University of Technology, China

It is proven that any Dembowski-Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used to explain some known planar Dembowski-Ostrom polynomials. A direct connection between a planar Dembowski-Ostrom polynomial and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. And all commutative semifields of order $3^5$ are classified.

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