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

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

This ebook constitutes the refereed complaints of the seventeenth overseas 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: jewelry, 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 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.

It can be shown that the highest possible dimension of a subspace of Mn,n (R) not containing any elements of rank 1 is directly related to the question of which k are possible. It has also been shown that subspaces consisting of all symmetric matrices, or all skewsymmetric matrices, are of similar importance to the problem of constructing embeddings into Euclidean space. Also, connections have been found between the embedding problem and the immersion problem, so the symmetric case has implications for the immersion problem.

The alphabet size 2 of the code as a function of the block length N is (N/ε2 )O(1/ε ) . The result of [8] also works in a more general setting called list recovery, which is defined next. Definition 2 (List Recovery). A code C ⊆ Σ n is said to be (ζ, l, L)-list recoverable if for every sequence of sets S1 , . . , Sn where each Si ⊆ Σ has at most l elements, the number of codewords c ∈ C for which ci ∈ Si for at least ζn positions i ∈ {1, 2, . . , n} is at most L. A code C ⊆ Σ n is said to (ζ, l)-list recoverable in polynomial time if it is (ζ, l, L(n))-list recoverable for some polynomially bounded function L(·), and moreover there is a polynomial time algorithm to find the at most L(n) codewords that are solutions to any (ζ, l, L(n))-list recovery instance.

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