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

This e-book constitutes the refereed complaints of the 4th foreign 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 conscientiously reviewed and chosen from 29 submissions. The papers are prepared in topical sections on coding concept and code-based cryptography, Boolean services, finite box mathematics, equations and services, and polynomial factorization and permutation polynomial.

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

Example text

U | gcd( Δ 4 , u) = 1 and 0 ≤ u < 4λ Δ gcd(λ,2) 4 , 2k n gcd(λ,2) } . therefore the result now follows directly from The following are direct applications of Theorem 3. Example 2. If q = 7, k = 2 and λ = 3, then Δ = 8 and N(7,2,3) = 3. In fact, if q = 7, k = 2 and λ = 3, then the family of cyclic codes C(a1 ,a2 ) described by Theorem 2 are C(2,26) , C(6,30) and C(10,34) . These three codes are 4-weight non-projective cyclic codes of length 24, dimension 4 and weight enumerator polynomial A(z) = 1 + 24z 6 + 216z 12 + 864z 18 + 1296z 24.

Assume N = ( q − 1)g. That is, we assume that the function field attains the Drinfeld-Vladut bound. Using codes CL (U = P1 + · · · + PN −1 , mQ) with g < m as outer codes one gets for all and k (see Section 2 for a discussion) Ω(k) X ⊆ F2 |X | = O , 3 k log(1/ ) . This result which is in the folklore is known as the AG-bound. 1 – Using Hermitian codes with m < g as outer codes one achieves [2] for ≥ k − 2 Ω(k) X ⊆ F2 |X | = O , k 2 log(1/ ) 5 4 . (3) This we call the BT-bound after the authors of [2], Ben-Aroya and Ta-Shma.

N }, (7) where λi ≤ g − 1 + i for i = 1, . . , g. This is a general result for Weierstrass semigroups and not particular for the Hermitian function field. Having described the Hermitian codes as affine variety codes we are now ready to introduce the combination of codes on which our construction of small-bias spaces rely. Consider the ideal 2 (2) 2 2 2 Iq2 := X1q+1 − Y1q − Y1 , X2q+1 − Y2q − Y2 , X1q − X1 , Y1q − Y1 , X2q − X2 , Y2q − Y2 and the corresponding variety (2) VFq2 (Iq2 ) = VFq2 (Iq2 ) × VFq2 (Iq2 ) = {Q1 , .

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