Arithmetic of Finite Fields: 4th International Workshop, by Florian Hess (auth.), Ferruh Özbudak, Francisco

By Florian Hess (auth.), Ferruh Özbudak, Francisco Rodríguez-Henríquez (eds.)

This publication constitutes the refereed complaints of the 4th overseas Workshop at the mathematics of Finite box, WAIFI 2012, held in Bochum, Germany, in July 2012. The thirteen revised complete papers and four invited talks awarded have been rigorously reviewed and chosen from 29 submissions. The papers are equipped in topical sections on coding thought and code-based cryptography, Boolean features, finite box mathematics, equations and services, and polynomial factorization and permutation polynomial.

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Extra info for Arithmetic of Finite Fields: 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012. Proceedings

Example text

A cyclic code C(a1 ,a2 ) belongs to the family of codes described by The- q −1 j orem 2, if a2 n ≡ 0 (mod q k − 1), gcd( Δ 2 , a2 ) = 2 and a1 = a2 q ± 2 , for q−1 j k some integer j with 1 ≤ q < q . Then a2 = λ u, for some integer u. If we supq−1 Δ pose that gcd( Δ 2 , λ ) > 2 then, clearly, gcd( 2 , a2 ) > 2. Therefore N(q,k,λ) = 0, q−1 Δ q−1 if gcd( 2 , λ ) > 2. Thus we will suppose that gcd( Δ 2 , λ ) ≤ 2. Now, since each one of the minimal polynomials ha1 (x) and ha2 (x) has exactly k differqk −1 Δ j ent conjugate roots, and since we have gcd( Δ 2 , a1 ) = gcd( 2 , a2 q ± 2 ) = Δ Δ Δ j j gcd( Δ , a q ± (q − 1)) = gcd( , a q ) = gcd( , a ) for any integer j, then 2 2 2 2 2 2 2 k k {a2 | a2 n ≡ 0 (mod q k − 1), gcd( Δ 2 , a2 ) = 2 and 0 ≤ a2 < (q − 1)} 2k q−1 , u) = 2 and 0 ≤ u < n} {u | gcd( Δ 2 λ .

Geil, S. Martin, and R. Matsumoto codes as described below. The easiest way to explain the combination is by using the language of affine variety codes [4] and we therefore start our investigations with a presentation of Hermitian codes as such. Definition 3. Given a monomial ordering ≺ and an ideal I ⊆ F[X1 , . . , Xm ] (here F is any field) the footprint is αm αm Δ≺ (I) := {X1α1 · · · Xm | X1α1 · · · Xm is not a leading monomial of any polynomial in I}. We have the following two useful results [3, Pro.

To estimate the dimension we make use of the characterization (7). The task is to estimate the number of (λ1 , λ2 )s that satisfies (q 3 − λ1 )(q 3 − λ2 ) ≥ δ. For this purpose we can replace Λ∗ with {g, g + 1, . . , q 3 − 1} ∪ {λn−g+1 , . . , λn }. A New Method for Constructing Small-Bias Spaces from Hermitian Codes 39 When estimating the dimension k(E(δ)) we shall furthermore ignore the elements in {λn−g+1 , . . , λn }. Writing T = q 3 − g we thereby get k(E(δ)) ≥ |{(i, j) | 0 ≤ i, j ≤ T − 1, (T − i)(T − j) ≥ δ}| T − Tδ ≥ 0 T − T δ−i djdi = T 2 − δ + ln .

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