By Florian Hess (auth.), Ferruh Özbudak, Francisco Rodríguez-Henríquez (eds.)
This ebook constitutes the refereed lawsuits 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 offered have been conscientiously reviewed and chosen from 29 submissions. The papers are prepared in topical sections on coding conception and code-based cryptography, Boolean services, finite box mathematics, equations and capabilities, and polynomial factorization and permutation polynomial.
Read Online or Download Arithmetic of Finite Fields: 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012. Proceedings PDF
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Extra resources for Arithmetic of Finite Fields: 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012. Proceedings
Then, the third identity of Pless (see [4, p. 259] for the general result), for C, is N wi2 Ai = [n(q − 1)(n(q − 1) + 1) − B1 (q + 2(n − 1)(q − 1)) + 2B2 ]q 2k−2 . (3) i=1 In the context of the previous identity, observe that a linear code is projective if and only if B1 and B2 are zero in (3). By keeping in mind this identity, we are now able to obtain the weight distribution of a family of non-irreducible cyclic codes. Theorem 2. Let q, k and Δ be as before. Considering our main assumption, we also take λ to be a divisor of q − 1 and define n = λΔ.
Therefore for all l ≥ 5 indeed (13) is better than (12). The situation is illustrated in Figure 2. Acknowledgments. The present work was done while Ryutaroh Matsumoto was visiting Aalborg University as a Velux Visiting Professor supported by the Villum Foundation. The authors gratefully acknowledge this support. The authors also gratefully acknowledge the support from the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography.
It turns out that for α < 1/2 the situation is rather complicated. We include the description here, as to our best knowledge, the details cannot be found in the literature. 34 O. Geil, S. Martin, and R. Matsumoto Fig. 1. Comparison of various constructions: First axis is α, second axis corresponds to logk (|X |) when k → ∞. 2 The AG-Bound Let q be a power of 2 and consider an algebraic function ﬁeld over Fq2 of genus g with at least N = (q − 1)g rational places. That is, the function ﬁeld attains the Drinfeld-Vladut bound.
Arithmetic of Finite Fields: 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012. Proceedings by Florian Hess (auth.), Ferruh Özbudak, Francisco Rodríguez-Henríquez (eds.)