Applied Algebra, Algebraic Algorithms and Error-Correcting by G. David Forney Jr. (auth.), Marc Fossorier, Hideki Imai,

By G. David Forney Jr. (auth.), Marc Fossorier, Hideki Imai, Shu Lin, Alain Poli (eds.)

This booklet constitutes the refereed court cases of the nineteenth foreign Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-13, held in Honolulu, Hawaii, united states in November 1999.
The forty two revised complete papers awarded including six invited survey papers have been rigorously reviewed and chosen from a complete of 86 submissions. The papers are equipped in sections on codes and iterative interpreting, mathematics, graphs and matrices, block codes, jewelry and fields, interpreting tools, code building, algebraic curves, cryptography, codes and interpreting, convolutional codes, designs, interpreting of block codes, modulation and codes, Gröbner bases and AG codes, and polynomials.

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Trivially, |G| − 1 ≤ L(G) ≤ 2|G| · (|G| − 1), and L(G) ≤ L2 (G). A theorem by Morgenstern [7] combined with the classical Schur relations yield [2]: L2 (G) > 14 |G| log |G|. Thus performing a DFT with only O(|G| log |G|) additions, subtractions, and scalar multiplications by small program constants is almost optimal in the L2 -model. This justifies the name Fast Fourier Transform. (For more details on lower complexity bounds, see Chapter 1 In the classical FFT algorithms these program constants are the so-called twiddle factors.

Gn , this cannot be done directly at level i. To this end we choose a suitable representation F ∈ Irrep(Gn , Tn ) whose restriction to CGi contains Di,1 as its first irreducible constituent. Then all what remains to do is to compute the first position of the diagonal matrix F (wi ), which equals Di,1 (wi ) = Di,1 (giγi ) = ζ γi . As F (wi ) = F (gi+1 )−γi+1 · · · F (gn )−γn · F (gn )αn · · · F (gi )αi · F (gn )βn · · · F (gi )βi is a product of monomial matrices and we are interested in only one entry of the final result, each factor F (gj ) causes only one addition in ZZ e .

All matrices Di,k (gl ), 1 ≤ i ≤ n, 1 ≤ k ≤ hi (hi denoting the n number of conjugacy classes of Gi ), 1 ≤ l ≤ i, is proportional to i=1 i · d1 (Gi ), n which is bounded from above by i=1 log(|Gi |) · d1 (Gi ). One can show that the number of operations of the algorithm is of this n magnitude O( i=1 log(|Gi |) · d1 (Gi )) with a moderate constant ≤ 20. In this sense the algorithm is nearly optimal. The following table shows the running times for some small supersolvable groups to construct all the matrices Di,k (gl ) as above.

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