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

ISBN-10: 3540772235

ISBN-13: 9783540772231

ISBN-10: 3540772243

ISBN-13: 9783540772248

This ebook 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 offered including 8 invited papers have been rigorously reviewed and chosen from sixty one submissions. one of the matters addressed are block codes, together with list-decoding algorithms; algebra and codes: jewelry, 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.

**Read Online or Download Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 17th International Symposium, AAECC-17, Bangalore, India, December 16-20, 2007. Proceedings PDF**

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

**Example text**

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 e = 0 subspace has no elements of rank n − 5, as shown above. What about the d = 0 subspace? This consists of forms Bc,e (x, y) = tr(c(x2 y + xy 2 ) + e(x8 y + xy 8 )).

Then we have dimK (R) ≤ k. We now present a result promised in the previous section. Theorem 2. Let L/K be a cyclic extension of degree n, n odd, with Galois group 3 generated by σ. Consider the set of bilinear forms tr(c(xσ y + xy σ ) + e(xσ y + 3 xy σ )) where c, e ∈ L. Then the ranks of these forms are n − 1 or n − 3. 3 3 Proof: Let Bc,e = tr(c(xσ y + xy σ ) + e(xσ y + xy σ )). By deﬁnition, 3 3 rad(Bc,e ) = {x ∈ L : tr(c(xσ y + xy σ ) + e(xσ y + xy σ )) = 0 ∀y ∈ L} = {x ∈ L : tr((cxσ + cσ −1 xσ −1 3 3 + eσ xσ + eσ −3 xσ −3 )y) = 0 ∀y ∈ L}.

1 Background Nothing in this section is new. We will use some motivating examples, which illustrate all the important ideas. In this section n is odd. For a ∈ L the function Ba (x, y) = tr(a(x2 y + xy 2 )) a symplectic bilinear form on L. The rank of Ba is n − wa where wa = dim rad(Ba ). By deﬁnition, rad(Ba ) = {x ∈ L : tr(a(x2 y + xy 2 )) = 0 ∀y ∈ L} n−1 = {x ∈ L : tr((ax2 + a2 n−1 x2 )y) = 0 ∀y ∈ L}. Since the trace form is nondegenerate, x is in rad(Ba ) if and only if ax2 + n−1 n−1 a2 x2 = 0.

### Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 17th International Symposium, AAECC-17, Bangalore, India, December 16-20, 2007. Proceedings by Venkatesan Guruswami (auth.), Serdar Boztaş, Hsiao-Feng (Francis) Lu (eds.)

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