Applied algebra, algebraic algorithms and error-correcting by Serdar Boztas, Hsiao-feng Lu

By Serdar Boztas, Hsiao-feng Lu

This ebook constitutes the refereed complaints 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 eight 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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The concatenated code, denoted by C = Cout ◦ Cin , is defined as follows. Let the rate of Cout be R and let the block lengths of Cout and Cin be N and n respectively. Define K = RN and r = k/n. The input to C is a vector m = m1 , . . , mK ∈ ([q]k )K . Let Cout (m) = x1 , . . , xN . The codeword in C corresponding to m is defined as follows C(m) = Cin (x1 ), Cin (x2 ), . . , Cin (xN ) . It is easy to check that C has rate rR, dimension kK and block length nN . Notice that to construct a q-ary code C we use another q-ary code Cin .

In differential topology, one important problem is the construction of immersions from real projective space Pn (R) into Euclidean space Rn+k . There are open problems dating from the early 1960s concerning the minimal possible k for which such an immersion exists. Let Mn,n (F ) denote the vector space of all n × n matrices over a field F . 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.

One might ask whether all near-bent functions arise in this way? The answer is no. Because, note that the function g is a bent function (namely f ) when restricted to the hyperplane y = 0. There are near-bent functions that are not bent when restricted to any hyperplane – we have checked this by computer for some Kasami-Welch near-bent functions, for example. Such near-bent functions cannot arise from this construction. 2 Going Down from a Bent Function Given a bent function on Vn , n even, we wish to consider restriction to a hyperplane to create a near-bent function in n − 1 variables.

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