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

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

This publication constitutes the refereed court cases 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 awarded including 8 invited papers have been conscientiously 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, diversifications, lattices; cryptography: cryptanalysis and complexity; computational algebra: algebraic algorithms and transforms; sequences and boolean functions.

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Extra resources 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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As a warm up, let us consider the case when s = m = 1. Note that for m = 1, we are interested in list decoding Reed-Solomon codes. More precisely, given the received word y = y0 , . . , yn−1 , we√are interested in all degree k polynomials i n − 1, f (X) such that for at least (1 + δ) R fraction of positions 0 f (γ i ) = yi . We now sketch the main ideas of the algorithms in [14,11]. The algorithms have two main steps: the first is an interpolation step and the second one is a root finding step. In the interpolation step, the list-decoding algorithm finds a bivariate polynomial Q(X, Y ) that fits the input.

2 This is well-defined on the quotient space L/ h . Since exactly one of a, a + h is in H, one of f (a), f (a + h) is 0 and the other is ±2(n+1)/2 . Therefore g(a) = ±2(n−1)/2 , so g is bent. Note that the proof did not require H to be a hyperplane; the argument only required that exactly one of a, a + h is in H. For the Kasami-Welch function k k k(x) = x4 −2 +1 with 3k ≡ ±1 (mod n), the support of k is the set of a ∈ L k with tr(a2 +1 ) = 1. , the trace 0 elements). k k In [3] the Kasami-Welch functions x4 −2 +1 are considered in greater detail.

Next, one could ask for subspaces where all ranks are n − 1 or n − 3. By [4], the maximum dimension for such a subspace is 2n. A function with spectrum {0, ±2(n+1)/2, ±2(n+3)/2 } should correspond to such a subspace, under the connection we have illustrated. Here is an example (from [4]). Consider the set of bilinear forms Bc,d (x, y) = tr(c(x2 y + xy 2 ) + d(x4 y + xy 4 )) over all c, d ∈ L. This set of bilinear forms is an F2 -vector space of dimension 2n. We claim that each nonzero form has rank n − 1 or n − 3.

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