By Wilfried Meidl, Arne Winterhof (auth.), Gary L. Mullen, Alain Poli, Henning Stichtenoth (eds.)

ISBN-10: 3540213244

ISBN-13: 9783540213246

ISBN-10: 3540246339

ISBN-13: 9783540246336

This booklet constitutes the completely refereed post-proceedings of the seventh overseas convention on Finite Fields and functions, Fq7, held in Toulouse, France, in could 2004.

The 19 revised complete papers provided have been conscientiously chosen from round 60 shows on the convention in the course of rounds of reviewing and revision. one of the issues addressed are Weierstrass semigroups, Galois jewelry, hyperelliptic curves, polynomial irreducibility, pseudorandom quantity sequences, permutation polynomials, random polynomials, matrices, functionality fields, ramified towers, BCH codes, cyclic codes, primitive polynomials, masking sequences, cyclic decompositions.

**Read or Download Finite Fields and Applications: 7th International Conference, Fq7, Toulouse, France, May 5-9, 2003. Revised Papers PDF**

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**New PDF release: Finite Fields and Applications: 7th International**

This booklet constitutes the completely refereed post-proceedings of the seventh foreign convention on Finite Fields and purposes, Fq7, held in Toulouse, France, in may well 2004. the nineteen revised complete papers provided have been rigorously chosen from round 60 shows on the convention in the course of rounds of reviewing and revision.

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**Extra info for Finite Fields and Applications: 7th International Conference, Fq7, Toulouse, France, May 5-9, 2003. Revised Papers**

**Example text**

A non right cyclic ring S ∈ GGR(D, pn ) may contain elements of maximal right order. According to Corollary 1 this is only possible if D is a right primitive semiﬁeld. The following result guarantees the existence of maximal right order elements in S when D is a right primitive semiﬁeld of odd characteristic. Proposition 2. Let S ∈ GGR(D, pn ) be a GGR with D a right primitive semiﬁeld of odd characteristic. Then S contains an element of maximal right order. Proof. e. ordr (w) = pd − 1 where |D| = pd , and take α ∈ S ∗ such that α = w.

Pd − 2}, since Scrit = RLα (e) there exists C2r ∈ RLα such that C2r (e) ≡2 e + pαr) . So there exists γ ∈ Γ = p d {0, e, α, α2) , . . , αp −2) } such that C2r (e) ≡3 e + pαr) + p2 γ. According to Lemma p 1 the congruence C2r (β) ≡2 β + pγβ holds for any element β ∈ Γ (for suitable p γβ ∈ Γ ). Let us prove by induction that for any natural number k ≥ 2 the following equations are true: (C2r )k (e) ≡3 e + kpαr) + kp2 γ + p k 2 p γαr) 2 and (C2r )k (αr) ) ≡2 αr) + kpγαr) . p Indeed: (C2r )2 (e) ≡3 C2r (e + pαr) + p2 γ) ≡3 e + 2pαr) + p2 (2γ + γαr) ) p p and (C2r )2 (αr) ) ≡2 C2r (αr) + pγαr) ) ≡2 αr) + 2pγαr) .

0)x↓ | µ(x) ∈ (Fi (x))} is equal to L. But this is the set of all possible elements in the k-th coordinate of the K-linear combinations of {D−t CDt x↓ }t∈IN , so dimK ({D−t CDt x↓ }t∈IN ) = d. As a corollary to this theorem we can give an answer to our problem of right-left cyclic top-associative GGR. Corollary 3. Let S ∈ GGR(GF (pd ), pn ) (n > 1) be a top-associative not associative GGR, char S = 22 or odd, and let α ∈ S ∗ be an element such that α is a primitive element of GF (pd ). Then S ∗ = RLα (e) if and only if Rα = Lα .

### Finite Fields and Applications: 7th International Conference, Fq7, Toulouse, France, May 5-9, 2003. Revised Papers by Wilfried Meidl, Arne Winterhof (auth.), Gary L. Mullen, Alain Poli, Henning Stichtenoth (eds.)

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