Henk Hollmann (Philips Research Laboratories)

will speak on

Linear recurring sequence subgroups and automorphisms of cyclic codes

Time: 4:00PM
Date: Mon 14th April 2008
Location: Mathematical Sciences Seminar Room [map]

Abstract: Let $q=p^r$ be a prime power, and let $f(x)=x^m-f_{m-1}x^{m-1}- cdots -f_1x-f_0$ be
an irreducible polynomial over the finite field $GF(q)$ of size $q$. A zero $xi$ of $f$
is called {em nonstandard/} if the recurrence relation
[ u_m=f_{m-1}u_{m-1} + cdots + f_1u_1+f_0u_0 ]
can generate the powers of $xi$ in a nontrivial way, that is, with $u_0=1$ and
$f(u_1)
eq 0$. In 2003, Brison and Nogueira asked for a characterisation of all
nonstandard cases in the case $m=2$, and solved this problem for $q$ a prime.
The problem is still open for $m=2$ and general $q$.

In this talk, we first relate this classification problem to the problem of determining
which cyclic codes over $GF(q)$ possess extra permutation automorphisms.

Then we discuss two classes of examples of nonstandard finite field elements. Finally, we use
the known classification of the subgroups of $PGL(2,q)$ in a first step towards showing that
these examples exhaust all possibilities in the case where $m=2$.

(This talk is part of the Algebra and Number Theory series.)

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