By James L. Massey (auth.), Serdar Boztaş, Igor E. Shparlinski (eds.)
The AAECC Symposia sequence used to be all started in 1983 through Alain Poli (Toulouse), who, including R. Desq, D. Lazard, and P. Camion, geared up the ?rst convention. initially the acronym AAECC intended “Applied Algebra and Error-Correcting Codes”. through the years its that means has shifted to “Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes”, re?ecting the growing to be significance of complexity in either interpreting algorithms and computational algebra. AAECC goals to inspire cross-fertilization among algebraic equipment and their purposes in computing and communications. The algebraic orientation is in the direction of ?nite ?elds, complexity, polynomials, and graphs. The purposes orientation is in the direction of either theoretical and useful error-correction coding, and, considering AAECC thirteen (Hawaii, 1999), in the direction of cryptography. AAECC used to be the ?rst symposium with papers connecting Gr¨obner bases with E-C codes. The stability among theoretical and functional is meant to shift frequently; at AAECC-14 the point of interest was once at the theoretical facet. the most matters lined have been: – Codes: iterative interpreting, interpreting equipment, block codes, code building. – Codes and algebra: algebraic curves, Gr¨obner bases, and AG codes. – Algebra: earrings and ?elds, polynomials. – Codes and combinatorics: graphs and matrices, designs, mathematics. – Cryptography. – Computational algebra: algebraic algorithms. – Sequences for communications.
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Additional info for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 14th International Symposium, AAECC-14 Melbourne, Australia, November 26–30, 2001 Proceedings
This is called the “glue” space. Often the component codes have minimum weight 4 and then one wants no additional weight 4’s in the glue space and also to have it large enough to get, together with the component codes, a self-dual code. The resulting code is then labeled by the labels of the component codes. Let d6 be the [6, 2, 4] code with the following generator matrix 111100 . Similarly d2m is a [2m, m − 1, 4] code. Thus 4 d6 is the label of 110011 a [24,12,4] Type I code. The component codes have dimension 8 and the “glue” space has dimension 4.
Des. 45 Pe Pe 10 −3 Quaternion 10 Orth. des. −3 10 Diagonal Quaternion −4 10 −4 10 −5 10 −5 10 0 5 10 15 20 SNR (dB) 25 30 35 0 5 10 15 20 25 30 35 SNR (dB) Fig. 1. Comparison of SL(2, F5 ) with orthogonal designs, diagonal codes, and the Quaternion group. The left picture is for one receiver antenna, while the second is for two receiver antennas. 45). Even though we now know all fpf finite groups, it may still be that there are fulldiversity space-time codes that are not groups themselves, but generate a finite group.
P mod (x − xn ) = p(x) as expected. The following deﬁnition summarizes the family of codes obtained this way. Deﬁnition 2 (Ideal error-correcting codes ).
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 14th International Symposium, AAECC-14 Melbourne, Australia, November 26–30, 2001 Proceedings by James L. Massey (auth.), Serdar Boztaş, Igor E. Shparlinski (eds.)