Abstract: We show how Grobner basis theory can be used in coding
theory, especially in the construction and decoding of linear codes.
A new method is given for construction of a large class of linear codes
that has a natural decoding algorithm. It works for any finite field
and any block length. The codes constructed include as special cases
many of the well known codes such as Reed-Solomon codes, Hermitian
codes and, more generally, all one-point algebraic geometry codes.
This method also allows us to construct random codes for which
our decoding algorithm performs reasonably well. Joint work with
Jeffrey B. Farr.

Let $K$ be a number field and $E/K$ an elliptic curve without
complex multiplication. A well-known theorem of Serre asserts that the
Galois group of $K(E[\ell])/K$ is as all of ${\rm GL}_2(\Z/\ell)$ for any
sufficiently large prime $\ell$. If we replace $E/K$ by a polarized abelian
variety $A/K$ with trivial endomorphism ring, then Serre later showed
that the Galois group of $K(A[\ell])/K$ is also as large as possible, for
all sufficiently large $\ell$, provided $\dim(A)$ is 2,6 or odd. We will
show how to prove a similar result for `most' $A$ and without any
restriction on $\dim(A)$.

Abstract:

In this talk, I will present a connection between designing low
correlation zone (LCZ) sequences and the results of correlation
of sequences with subfield decompositions. This results
in low correlation zone signal sets with huge sizes over three
different alphabetic sets: finite field of size $q$, integer
residue ring modulo $q$, and the subset in the complex field which
consists of powers of a primitive $q$-th root of unity. A connection between these
constructions and ``completely
non-cyclic'' Hadamard matrices will be shown. I will also provide some open problems
along this direction.

Joint work with Solomon W. Golomb and Hongyeop Song.

Brumer's conjecture states that Stickelberger elements combining values of L-functions at s=0 for an abelian extension of number fields E/F should annihilate the ideal class group of E when it is considered as module over the appropriate group ring. In some cases, an ideal obtained from these Stickelberger elements has been shown to equal a Fitting ideal connected with the ideal class group. We consider the analog of this at s=-1, in which the class group is replaced by the tame kernel, which we will define. For a field extension of degree 2, we show that there is an exact equality etween the Fitting ideal of the tame kernel and the most natural higher Stickelberger ideal; the 2-part of this equality is conditional on the Birch-Tate conjecture.