.
DEPARTMENT
of
MATHEMATICS
The Galois Variance of Constacyclic Codes
Professor Tom Blackford
Western Illinois University
Abstract: Some of the earliest error-correcting codes (used in
the encoding and transmission of data, pictures, and sound)
were linear codes that were constacyclic. If Fq is a finite field,
and λ is a nonzero element of Fq , a constacyclic code over Fq
of length n is a subspace C of (Fq )n with the property that
(c0 , c1 , . . . , cn−1 ) ∈ C =⇒ (λcn−1 , c0 , · · · , cn−2 ) ∈ C,
so that C is invariant under a λ-cyclic shift of coordinates.
Constacyclic codes have a rich algebraic structure, and can
be viewed as ideals of a quotient ring of Fq [x]. They can
be classified by generator polynomials and defining sets. We
will look at the images of constacyclic codes over Fqm under
the Galois group Gal(Fqm /Fq ). In particular, we will examine their restriction and trace subcodes, as well as determine
which codes are Galois invariant and which codes are Galois
disjoint. We will also look at an interesting class of completely
Galois disjoint codes.
Thursday,
April 7, 2016
3:45 p.m.
Morgan Hall 204
Refreshments
will be served at 3:30
p.m.