Applied Algebra, Algebraic Algorithms and Error-Correcting by Peter Beelen, Diego Ruano (auth.), Maria Bras-Amorós, Tom

By Peter Beelen, Diego Ruano (auth.), Maria Bras-Amorós, Tom Høholdt (eds.)

This ebook constitutes the refereed complaints of the 18th overseas Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-18, held in Tarragona, Spain, in June 2009.

The 22 revised complete papers offered including 7 prolonged absstracts have been conscientiously reviewed and chosen from 50 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: 18th International Symposium, AAECC-18 2009, Tarragona, Spain, June 8-12, 2009. Proceedings

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F, ϕ are coprime. 4. f is monic 2 d−1 2 Consider the following algorithm: Initialize: r−2 = xn − 1, f−2 = 0, ϕ−2 = −1, ¯ f−1 = 1, ϕ−1 = 0, r−1 = S, i = −1. while deg(ri ) > n − d+1 : 2 qi = Quotient(ri−2 , ri−1 ) ri = Remainder(ri−2 , ri−1 ) ϕi = ϕi−2 − qi ϕi−1 fi = fi−2 − qi fi−1 end while Return fi /LeadingCoefficient(fi ), ϕi /LeadingCoefficient(fi ) The polynomial ϕ has been initialized to negative 1 because we want that fi , ϕi satisfy at each step fi S¯ − ϕi (xn − 1) = ri . The next theorem verifies the algorithm.

We say that H is acute if it satisfies (I); H is near-acute if it satisfies (II) or (III). Notice that (I) implies (II). Define the function ι on H by ι(ρi ) = i. In [8, Thm. 11] it is shown that o(H) = min{ι(2d1 + 1), ι(c0 + c1 − 1)} provided that H is near-acute (*). See also [10,11] for more information. Proposition 5. Let H be a sparse semigroup of genus g with 2 g ≥ 3g then H is acute and o(H) = 2 g − g = 3g − 2K. g = 2g − K. If Proof. Notation as in equation (4). From Theorem 1, c0 = g +1, c1 = d1 = g −1 and c2 = d2 = g − 3.

The quaternary linear Reed-Muller codes RMs (r, m) of length 2m−1 , for m ≥ 1, 0 ≤ r ≤ m and 0 ≤ s ≤ m−1 , in these new families satisfy that the corresponding Z4 -linear 2 codes have the same parameters and properties (length, dimension, minimum distance, inclusion and duality relationship) as the binary linear codes in the wellknown RM family. In the binary case, there is only one family. In contrast, in the families, which will be distinguished quaternary case, for each m there are m+1 2 }. using subindexes s from the set {0, .

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