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A family of asymptotically good lattices having a lattice in each dimension. (English) Zbl 1200.11047

Even though reasonable bounds about the best density that a lattice packing can achieve are known, namely \(\frac{1}{n} \log_2 \Delta \geq -1\) where \(n\) is the dimension and \(\Delta\) the density, up to this date, no good family of explicit lattices meeting that bound has been constructed. This article exhibits a family of lattices such that \(\frac{1}{n} \log_2 \Delta \geq -\frac{1}{2} \log_2 \log_2 n + \kappa\) with \(\kappa = -0.6395...\) for any dimension \(n\). Constructions existed until then for only specific values of \(n\). To do so, the authors extend the family of Craig lattices. Their lattices are obtained in a subfield of the cyclotomic field \(\mathbb{Q}(\zeta_q)\), where \(q\) is the smallest prime congruent to 1 modulo \(n\), from a well chosen integral ideal, namely some power of the restriction to the subfield of the ideal spanned by \(1-\zeta_q\).

MSC:

11H31 Lattice packing and covering (number-theoretic aspects)
11R18 Cyclotomic extensions
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References:

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