By Peter Beelen, Diego Ruano (auth.), Maria Bras-Amorós, Tom Høholdt (eds.)
This publication constitutes the refereed lawsuits of the 18th foreign Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-18, held in Tarragona, Spain, in June 2009.
The 22 revised complete papers provided 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: jewelry and fields, polynomials, diversifications, lattices; cryptography: cryptanalysis and complexity; computational algebra: algebraic algorithms and transforms; sequences and boolean functions.
Read or Download Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 18th International Symposium, AAECC-18 2009, Tarragona, Spain, June 8-12, 2009. Proceedings PDF
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Extra info for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 18th International Symposium, AAECC-18 2009, Tarragona, Spain, June 8-12, 2009. Proceedings
The code RMs (0, m) is deﬁned as the repetition code with only the all-zero and all-two vectors. The code RMs (r, m) with r ≥ m is deﬁned . For m = 1, there is only one family with s = 0, and as the whole space Zm−1 4 in this family there are only the zero, repetition and universe codes for r < 0, r = 0 and r ≥ 1, respectively. In this case, the generator matrix of RM0 (0, 1) is G0(0,1) = 2 and the generator matrix of RM0 (1, 1) is G0(1,1) = 1 . For any m ≥ 2, given RMs (r, m − 1) and RMs (r − 1, m − 1) codes, where 0 ≤ s ≤ m−2 , the RMs (r, m) code can be constructed in a recursive way 2 using the Plotkin construction given by Deﬁnition 2 as follows: RMs (r, m) = PC(RMs (r, m − 1), RMs (r − 1, m − 1)).
124 (= g), dDK is the maximum lower bound for a two-point code deﬁned with C = CP P + CQ Q, and CQ gives values for which the maximum is achieved. Suppressed are divisors that deﬁne subcodes with the same minimum distance as already listed codes. Exchanging P and Q gives a similar code and listed are only divisors with CQ (mod m) ≤ CP (mod m). The last columns give the amount by which dDK exceeds similarly deﬁned maximum lower bounds for dDP and dB , respectively. 4], and the order bounds dB0 , dB , dABZ , dDP , dDK .
These matrices will satisfy at given stages of the algorithm (exactly when deg(Ri ) ˜ i )) that Ri , Fi , Ψi are the remainder and the intermediate B´ezout coeﬃ< deg(R ˜ i , F˜i , Ψ˜i cients in one of the steps of the original Euclidean algorithm and that R are respectively the remainder and the intermediate B´ezout coeﬃcients previous to Ri , Fi , Ψi . 1 Making Remainders Monic ˜ i monic in all steps. This makes it easier to A useful modiﬁcation is to keep R ˜i ). compute the µ’s. It is enough to force this every time we get deg(Ri ) < deg(R ˜ i ), then R ˜ i stays the same and so it remains monic.
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 18th International Symposium, AAECC-18 2009, Tarragona, Spain, June 8-12, 2009. Proceedings by Peter Beelen, Diego Ruano (auth.), Maria Bras-Amorós, Tom Høholdt (eds.)