Linear recurring sequence subgroups and automorphisms of cyclic codes

Speaker: Henk Hollmann (Philips Research Laboratories)

Time: 4:00PM

Date: Mon 14th April 2008

Location: Mathematical Sciences Seminar Room

Let q=pr be a prime power, and let f(x)=xm−fm−1xm−1−cdots−f1x−f0 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 u0=1 and f(u1)eq0. 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/Claude Shannon Institute series.)