Graham Norton
(Brisbane)
Cyclic codes and Groebner bases over a principal ideal ring
Thursday 7th June, 3-4pm
Eastern Avenue Tutorial Room 404
Let p be a prime and let n,k be integers, gcd(p,n)=1 and k>1. Calderbank and
Sloane characterised cyclic codes of length n over the integers modulo p^k in
a form which intuitively suggested a 'minimal strong Groebner basis (SGB)'
over the integers modulo p^k.
Let D be a principal ideal domain. The structure of a minimal SGB for an ideal
of D[x_1,...,x_n] is due to Becker and Weisspfenning, Lazard and others.
We outline an effective theory of minimal SGB's for ideals of R[x_1,...,x_n],
where R is an arbitrary principal ideal ring (e.g. the integers modulo p^k),
generalising the PID case.
We then characterise cyclic codes of arbitrary length over R using minimal
SGB's. This yields the result of Calderbank and Sloane as a special case.
This is joint work with Ana Salagean, Nottingham Trent University, UK.
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