# 20130124

## 九州大学 代数・組合せ数学 日韓合同ワークショップ †

**～ Japan-Korea Workshop on Algebra and Combinatorics ～**

The 11th Japan-Korea Workshop on Algebra and Combinatorics will take place at Fukuoka City in Japan, from 24-25, January 2013. The workshop is held once a year, alternatively in Japan and in Korea. It is intended to provide researchers of both countries, especially young researchers including graduate students, with opportunities to exchange rather informal information of ongoing studies in the area.

Further information is available from the organizers below.

### Organizers †

- Eiichi Bannai (Shanghai Jiao Tong University)
- Jung Rae Cho (Pusan National University)
- Mitsugu Hirasaka (Pusan National University)
- Tatsuro Ito (Kanazawa University)
- Hyun Kwang Kim (POSTECH)
- Jack Koolen (POSTECH)
- Yoshihiro Mizoguchi (Kyushu University),
- Akihiro Munemasa (Tohoku University)

### Supported by †

Global COE Program "Education and Research Hub for Mathematics-for-Industry"

### Date †

- January 24, 2013. 10:00-18:00
- January 25, 2013. 10:00-18:00

### Location †

- Seminar Room 2 in ACROS Fukuoka Foundation 1-1-1 Tenjin, Chuo-ku, Fukuoka City, 810-0001 (see http://www.acros.or.jp/english/access/ )

### Accommodation †

The place of the workshop is located at Tenjin area which is in the center of Fukuoka City and closed to Nakasu and Hakata. You can find many hotels around Tenjin, Nakasu or Hakata (see http://www.booking.com/ ).

### Satellite Seminar †

In 26th of January a workshop entitled as Hakata Workshop 2012 on "Combinatorics and its Applications" will be held at Reference Rental Meeting Room (see http://www.re-rental.com/index.html) in Hakata area.

### Program †

#### January 24 (Thursday) at Room 2 in ACROS Fukuoka †

- Openning Speech Yoshihiro Mizoguchi
- Chair: Akihiro Munemasa
- 10:00-10:45 Ferenc Szöllősi(Tohoku University)
- Modular combinatorial designs and Hadamard matrices modulo 5

- 11:00-11:45 Gary Greaves (Tohoku University)
- Minimal polynomials of integer symmetric matrices

- Chair: Jung Rae Cho
- 13:30-14:15 Ryuzaburo Noda
- Some bounds for the number of Blocks

- 14:30-15:15 Takao Komatsu (Hirosaki University)
- Poly-Cauchy numbers and their combinatorial properties

- Chair: Hyun Kwang Kim (POSTECH)
- 15:45-16:30 Jong Yoon Hyun (Ewha Womans University)
- A new family of strongly regular graphs using weakly regular
*p*-ary bent functions

- A new family of strongly regular graphs using weakly regular
- 16:45-17:30 Phan Thanh Toan (POSTECH) Counting methods and upper bounds on sizes of codes

#### January 25 (Friday) at at Room 2 in ACROS Fukuoka †

- Chair: Eiichi Bannai
- 10:00-10:45 Xiao-Dong Zhang (Shanghai Jiao Tong University)
- The Dirichlet eigenvalues of graphs and the Faber-Krahn Inequality

- 11:00-11:45 Shun'ichi Yokoyama (Kyushu University / JST CREST)
- Developments in Computer Algebra Research: the Next Generation

- Chair: Tatsuro Ito
- 13:30-14:15 Joon Yop Lee (POSTECH)
- Multidimensional matrices uniquely recovered by their lines

- 14:30-15:15 Satoshi Tsujimoto (Kyoto University)
- Dunkl shift operators and Bannai-Ito polynomials

- Chair: Jack Koolen
- 15:45-16:30 Andreas Holmsen (KAIST)
- Generalized wiring diagrams, realization spaces, and configurations of convex sets

- 16:45-17:30 Jon-Lark Kim (Sogang University)
- A new class of error-correcting codes for Boolean masking of cryptographic computations

- Closing Speech Mitsugu Hirasaka

### Abstract †

#### Gary Greaves (Tohoku University) †

- Title: Minimal polynomials of integer symmetric matrices
- Abstract:

The minimal polynomial of an integer symmetric matrix must be monic and must have integer coefficients and distinct real zeros. In this talk we will discuss advances about the extent to which these necessary condi- tions are sufficient and we will present some recent classifications of integer symmetric matrices having their spectral spread less than 4.

#### Jong Yoon Hyun (Ewha Womans University) †

- Title: A new family of strongly regular graphs using weakly regular
*p*-ary bent functions - Abstract:

We construct strongly regular graphs and association schemes by
using the weakly regular *p*-ary bent functions (from
to
), where *p* is a prime number. We obtain some useful sign function formulae for those functions, and these formulae are used for development of our main result. In fact, we find a new family of strongly regular graphs.

#### Andreas Holmsen (KAIST) †

Xiaojun Chen (The Hong Kong Polytechnic University)

- Title: Generalized wiring diagrams, realization spaces, and configurations of convex sets
- Abstract:

Wiring diagrams were introduced by J.Goodman in 1980 as a com- binatorial encoding of planar point configurations. Together with M.Dobbins and A.Hubard, we generalize this and consider the "wiring monoid" gener- ated by simple the simple transpositions. I will characterize certain highly regular substructures of the wiring monoid and show how this resolves a con- jecture of Pach and Toth in geometric Ramsey theory, by using a geometric representation theorem for generalized wiring diagrams. This also gives rise to the notion of Realization spaces for which I will describe our Universality result.

#### Jon-Lark Kim (Sogang University) †

Title: A new class of error-correcting codes for Boolean masking of cryptographic computations Abstract:

We introduce a new class of rate one half binary codes: complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this talk we investigate this new class of codes: we give optimal or best known CIS codes of length ｡ 132. We derive general constructions based on cyclic codes, double circulant codes, and 2-class association schemes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths up to 12 by the building up construction. Some nonlinear S-boxes are constructed by using Z4-codes, based on the notion of dual distance of an unrestricted code. This is a joint work with C. Carlet, P. Gaborit, and P. Sole.

#### Takao Komatsu (Hirosaki University) †

Title: Poly-Cauchy numbers and their combinatorial properties Abstract: We introduce some generalized poly-Cauchy numbers and show some com- binatorial properties. (to be updated)

#### Joon Yop Lee (POSTECH) †

- Title: Multidimensional matrices uniquely recovered by their lines
- Abstract:

We provide a method to determine if a *q*-ary multidimensional
matrix is lonesum or not by using properties of linesums of lonesum mul-
tidimensional matrices. In particular, we establish a graphic method that
uses edge-colored graphs to determine if a binary multidimensional matrix is
lonesum or not. We also provide two methods to determine if a *q*-ary mul-
tidimensional matrix is lonestructure or not. The first one uses properties
of line structures of lonestructure multidimensional matrices and the second
one uses edge-colored directed multigraphs.

#### Ryuzaburo Noda †

- Title: Some upper bound for the number of blocks in block designs
- Abstract:

Definition. A design *D* = (*, B*) is called a design with parameters
set (*v, k, d*) or simply a (*v, k, d*) design if |*Ω*| = *v*, block size=*k* and the
maximal intersection number of two blocks=*d*.

We do not assume that *D* is a *t*-design, though it proves to be a certain t-
design if some natural upper bound for the number of blocks of *D* is achieved.
The following lemma is simple but fundamental.

Lemma. Let *D* be a (*v, k, d*) design. Then for any point subset *X* of size
*d + 2i - 1*, where *i* is an arbitrary integer, there exists at most one block *B*
such that |*X* ⋂ *B*| is larger than or equal to *d + i*.

In [2] and [3] ,by making use of the above lemma,the natural upper bound
for the number of blocks in a (*v, k, d*) design is given and it is proved in the
case *i = 2* that the bound is achieved if and only if it is a *t*-design for a
certain t which is the zero of some polynomial of t with coefficients from *v*,
*k* and *d*. First I will show the results in [2],[3] and then show some recent
developments .The relation with perfect *e*-codes in Johnson scheme is also
shown.

[1]P.Hauck, Eine Charakterisierungdes Steiner systems J. Comb. Theory, Ser.A, 32(1982)

[2]R.Noda, Some Bounds for the number of Blocks, Europ. J. Combinatorics, 22 (2001) 91-94

[3]R.Noda, Some Bounds for the number of Blocks II, Europ. J. Combinatorics, 22 (2001) 95-100

[4] T.Etzion, Perfect constant-weight codes IEE Transactions on Information Theory Vol.50, No9, September, 2004

#### Ferenc Szöllősi(Tohoku University) †

- Title: Modular combinatorial designs and Hadamard matrices modulo 5
- Abstract:

Let *m > 2* be an integer. An *m*-modular Hadamard matrix
of
order *n* is an *n × n* matrix with (*-1, 1*) entries such that
(*mod m*).
In this talk we report on our recent advances regarding the existence of mod-
ular Hadamard matrices. By introducing a new concept, called m-modular
symmetric designs, we were able to give a complete classification of 5-modular
Hadamard matrices. This is a joint work with Prof. Moon Ho Lee.

#### Phan Thanh Toan (POSTECH) †

- Title: Counting methods and upper bounds on sizes of codes
- Abstract:

We study error-correcting codes over
, where *q ≧ 2* is an integer.
We prove (and reprove) several upper bounds on sizes of codes by using
counting methods. Explicit new upper bounds are given.

#### Satoshi Tsujimoto (Kyoto University) †

- Title: Dunkl shift operators and Bannai-Ito polynomials
- Abstract:

We consider the operator which contains the first order shift operator and the reflac- tion operator, that we call Dunkl shift operator. Then the Bannai-Ito poly- nomials can be introduced as a sequence of the polynomial eigenfunctions of the Dunkl shift operator. We will discuss various aspects of the Bannai-Ito polynomials from the viewpoint of classical orthogonal polynomials.

#### Shun'ichi Yokoyama (Kyushu University / JST CREST) †

- Title: Developments in Computer Algebra Research: the Next Generation
- Abstract:

Producing computer algebra systems and related services (almost all are non-commercial) has been of great value in mathematical research. Recently, high-quality several web-based software development environments are born. They are highly interactive and much more focused on math al- gorithms, education, and pure mathematics research (e.g. algebra and com- binatorics) of course. In this talk, we introduce some of CAS-development projects that are currently going on.

#### Xiao-Dong Zhang (Shanghai Jiao Tong University) †

- Title: The Dirichlet eigenvalues of graphs and the Faber-Krahn Inequality
- Abstract:

The Faber-Krahn inequality in the Riemannian manifolds states that the ball has minimal first Dirichlet eigenvalue among all bounded do- mains with the fixed volume in . In this talk, we will introduce the concept of a "graph with boundary " and formulated the Dirichlet eigenvalue problem for graphs. Then survey some new results and progress on the Faber-Krahn inequality of graphs and the first Dirichlet eigenvalues of graphs.

Joint work with Congpei An, Andreas Frommer, Bruno Lang, Ian Sloan and Womersley.

**References**