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. 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.