Maubach, S.
Polynomial automorphisms over finite fields
(pp. 343350)
A B S T R A C T S
DIVISIBLE CODES  A SURVEY
Harold N. Ward
hnw@virginia.edu
2000 Mathematics Subject Classification: 94B05.
Key words:
divisible code, theorem of Ax, ReedMuller code, group algebra,
DelsarteMcEliece theorem, polarization, selfdual code, GleasonPierce theorem,
Griesmer bound, optimal code.
This paper surveys parts of the study of divisibility properties of codes.
The survey begins with the motivating background involving polynomials over
finite fields. Then it presents recent results on bounds and applications
to optimal codes.
ON PROJECTIVE PLANE OF ORDER 13 WITH A FROBENIUS GROUP
OF ORDER 39 AS A COLLINEATION GROUP
Razim Hoxha
razimhoxha@yahoo.com
2000 Mathematics Subject Classification: 05B25.
Key words:
projective plane, collineation group, orbit structure.
One of the most outstanding
problems in combinatorial mathematics and geometry is
the problem of existence of finite projective planes
whose order is not a prime power.
NEW BINARY [70,35,12] SELFDUAL ABD BINARY [72,36,12]
SELFDUAL DOUBLYEVEN CODES
Radinka Dontcheva
r.a.doncheva@its.tudelft.nl
2000 Mathematics Subject Classification: 94B05, 94B60.
Key words:
Automorphisms, selfdual codes, weight enumerators.
In this paper we prove that up to equivalence there exist 158
binary [70,35,12] selfdual and 119 binary [72,36,12]
selfdual doublyeven codes all of which have an automorphism
of order 23 and we present their construction. All these codes are
new.
FURTHER GENERALIZATION OF KOBAYASHI'S GAMMA FUNCTION
L. Galue,
G. Alobaidi
alobaidi@math.uregina.ca,
S. L. Kalla
kalla@mcs.sci.kuniv.edu.kw
2000 Mathematics Subject Classification:
33B15, 33C15, 33B20, 41A60.
Key words:
Kobayashi's gamma function,
hypergeometric functions, asymptotic formulas.
In this paper, we introduce a further generalization of the gamma
function involving Gauss hypergeometric function _{2}F_{1}(a,b;c;z)
by means of:
(1)
D 
æ ç
è


a,b,c,p
u,v,d


ö ÷
ø

= v^{a} 
ó õ

¥
0

t^{u1} 
æ ç
è

1 
t
v


ö ÷
ø

d 1

e^{pt}_{2}F_{1} 
æ ç
è

a,b;c; 
t
v


ö ÷
ø

dt 

where Re u > 0, Re p > 0, argv  < p. This reduces to Kobayashi's
[7] generalized gamma function when
d = 1, p = 1 and b = c.
Also, it reduces to a function defined by AlMusallam and Kalla
[2, 3] when
d = 1. The generalized incomplete and the
complementary incomplete functions associated with
D([a,b,c,p  ( u,v,d)] ) are also introduced. For these
functions we obtain some properties and recurrence relations satisfied by
them and we establish asymptotic series expansions for each of them.
POLYNOMIALS OF PELLIAN TYPE AND CONTINUED FRACTIONS
R. A. Mollin
ramollin@math.ucalgary.ca
2000 Mathematics Subject Classification:
11A55, 11R11.
Key words:
continued fractions, Pell's Equation, period length.
We investigate infinite families of integral quadratic polynomials
{f_{k}(X)}_{k Î N} and show that, for a fixed k Î N
and arbitrary X Î N, the period length of the simple continued fraction
expansion of
^{}Ö{f_{k}(X)}
is constant. Furthermore, we show
that the period lengths of Ö{f_{k}(X)} go to infinity with k.
For each member of the families involved, we show how to determine,
in an easy fashion, the fundamental unit of the underlying
quadratic field. We also demonstrate how the simple continued
fraction expansion of Ö{f_{k}(X)} is related to that of
ÖC, where f_{k}(X) = a_{k}X^{2}+b_{k}X+C. This
continues work in [1][4].
POLYNOMIAL AUTOMORPHISMS OVER FINITE FIELDS
Stefan Maubach
stefanm@sci.kun.nl
2000 Mathematics Subject Classification:
14R10, 14R15, 11C08, 20B27, 20B25, 20B15, 20B99.
Key words:
polynomial automorphisms, tame automorphisms, affine spaces over
finite fields, primitive groups.
It is shown that the invertible polynomial maps over a finite field
F_{q}, if looked at as bijections F_{q}^{n}® F_{q}^{n},
give all possible
bijections in the case q = 2, or q = p^{r} where p > 2. In the case q = 2^{r}
where r > 1 it is shown that the tame subgroup of the invertible
polynomial maps gives only the even bijections, i.e. only half the bijections.
As a consequence it is shown that a set S Ì F_{q}^{n} can be a zero set
of a coordinate if and only if #S = q^{n1}.
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