Applied Algebra, Algebraic Algorithms and Error-Correcting by Venkatesan Guruswami (auth.), Serdar Boztaş, Hsiao-Feng

By Venkatesan Guruswami (auth.), Serdar Boztaş, Hsiao-Feng (Francis) Lu (eds.)

This publication constitutes the refereed lawsuits of the seventeenth foreign Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-17, held in Bangalore, India, in December 2007.

The 33 revised complete papers provided including 8 invited papers have been rigorously reviewed and chosen from sixty one submissions. one of the topics addressed are block codes, together with list-decoding algorithms; algebra and codes: earrings, fields, algebraic geometry codes; algebra: earrings and fields, polynomials, variations, lattices; cryptography: cryptanalysis and complexity; computational algebra: algebraic algorithms and transforms; sequences and boolean functions.

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Extra info for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 17th International Symposium, AAECC-17, Bangalore, India, December 16-20, 2007. Proceedings

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First we assume that H does not contain any Ha . Then each Ha is a hyperplane in Ha . The inner summation is 0 unless λ vanishes on Ha , and there are precisely two such a for any λ, as λ = 0. So f |H (λ) = ±2t−1 ± 2t−1 which is 0 or ±2t . Secondly, assume that H does contain one of the Ha , say H . ) Then the inner sum will be 0 unless λ is the unique linear functional whose kernel is H . Thus the value of f |H (λ) in this case is (−1)g(H ) 2t . We thank John Dillon for discussions about these functions.

I i k The same argument repeated for the forms tr( i=1 ci (x2 y + xy 2 )) will give kn-dimensional subspaces of matrices of ranks n − 1, n − 3, ... n − 2k + 1. This recovers a result of Delsarte and Goethals [4], which also appears in [6]. For example, in the 3n-dimensional space of forms tr(c(x2 y + xy 2 ) + d(x4 y + xy 4 ) + e(x8 y + xy 8 )) 32 G. McGuire all nonzero elements have rank n − 1, n − 3 or n − 5. This 3n-dimensional subspace contains three obvious 2n-dimensional subspaces, consisting of all elements where one of c, d, e is 0.

The construction of the P Sap bent functions starts with a balanced function g : K −→ F2 where K = F2t . Dillon’s result states that the function f (x, y) = n g(xy 2 −2 ) is bent on K × K (actually the result is more general, concerning n partial spread bent functions). Note that g(xy 2 −2 ) = g(x/y) if y = 0. In K × K let Ha denote the line {(x, ax) : x ∈ K} and let H∞ = {(0, y) : y ∈ K}. These 2t + 1 lines intersect pairwise in (0, 0) and partition K × K. The linear span of any two of these lines is K × K.

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