By James L. Massey (auth.), Serdar Boztaş, Igor E. Shparlinski (eds.)
The AAECC Symposia sequence used to be began in 1983 by way of 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 becoming significance of complexity in either interpreting algorithms and computational algebra. AAECC goals to motivate cross-fertilization among algebraic equipment and their functions 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 functional error-correction coding, and, due to the fact that AAECC thirteen (Hawaii, 1999), in the direction of cryptography. AAECC was once the ?rst symposium with papers connecting Gr¨obner bases with E-C codes. The stability among theoretical and functional is meant to shift usually; at AAECC-14 the focal point used to be at the theoretical aspect. the most topics coated have been: – Codes: iterative interpreting, interpreting tools, 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 resources for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 14th International Symposium, AAECC-14 Melbourne, Australia, November 26–30, 2001 Proceedings
Since p is odd, the multiplicative group of But σ = σ , so Z/(pn−1 Z) is cyclic and generated by an element, say ω. This shows that is in the n−3 subgroup generated by ω p (p−1) , which shows that ≡ 1 mod pn−2 . As a result, p divides, and p2 does not divide ( p − 1)/( − 1). To finish the proof, it suffices to show that p divides k. Consider R(τ p ). This is a diagonal matrix with all diagonal entries equal to η k . On the other hand, R(τ p ) = R(σ k ). p−1 The latter is a diagonal matrix with diagonal entries η k , η k , .
Shokrollahi M the arithmetic-geometric-mean inequality we know that i=1 |ρi |1/M ≤ ||A||/M . This shows that | det(A − B)|1/M ≤ ||A − B||/M , which implies that | det(A − B)| is upper bounded by the distance of A and B on the sphere. Hence, the maximum minimum diversity distance of a set of three matrices is upper bounded by (half) √ the maximum minimum distance of a set of three points on the sphere. The latter is √ 3/2. This shows √ that A(M, √ 3) ≤ 3/2. The proof of the lower bound A(M, 3) ≥ 3/2 follows from A(1, 3) = 3/2 and A(1, 3) ≤ A(M, 3) by Proposition 1(4).
135). The right hand side of the above inequality is, for large L, smaller than A(2, L), which shows that the construction above is worse than the trivial bound. However, this is due to the construction of the spherical code given in . Better constructions will lead to better results. Open Problem 5 Are there other ways to use representations of compact Lie groups to derive good space-time codes in other dimensions? We will discuss in the next section finite space-time group codes that are not fully diverse.
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.)