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A Mass Formula For Unimodular Lattices With No Roots
, 2002
"... We derive a mass formula for ndimensional unimodular lattices having any prescribed root system. We use Katsurada's formula for the Fourier coe#cients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32dimensional lattices and odd unimodular lattices ..."
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We derive a mass formula for ndimensional unimodular lattices having any prescribed root system. We use Katsurada's formula for the Fourier coe#cients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32dimensional lattices and odd unimodular lattices of dimension n 30. In particular, we find the mass of even unimodular 32dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension n 30, verifying Bacher and Venkov's enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those a#orded by the MinkowskiSiegel mass constants. 1.
Classification of Extremal and sExtremal Binary SelfDual Codes of Length 38
, 2013
"... Abstract—In this paper we classify all extremal and sextremal binary selfdual codes of length 38. There are exactly 2744 extremal [38; 19; 8] selfdual codes, two sextremal [38; 19; 6] codes, and 1730 sextremal [38; 19; 8] codes. We obtain our results from the use of a recursive algorithm used i ..."
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Abstract—In this paper we classify all extremal and sextremal binary selfdual codes of length 38. There are exactly 2744 extremal [38; 19; 8] selfdual codes, two sextremal [38; 19; 6] codes, and 1730 sextremal [38; 19; 8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal selfdual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of sextremal [38; 19; 6] codes permits to achieve the classification of all sextremal codes with d =6. Index Terms—Classification, extremal, recursive construction, selfdual codes,extremal, shadow. I.
Sextremal strongly modular lattices
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
"... Sextremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the sextremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an sextremal lattices can be bounded by the t ..."
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Sextremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the sextremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an sextremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many sextremal strongly modular lattices of even minimum.
Strongly modular lattices with long shadow
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX 16 (2004), 187–196
, 2004
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selfdual codes of length 38
, 1111
"... Classification of extremal and sextremal binary ..."
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Shadows of Odd Unimodular Lattices
"... This is an exposition of our recent joint paper with G. Nebe: G. Nebe and B. Venkov, “Unimodular lattices with long shadow. ” The paper is submitted to Journal of Number Theory. Full text can be found on the web page of G. Nebe. After Elkies and Gaulter, we study odd unimodular lattice $\Lambda $ in ..."
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This is an exposition of our recent joint paper with G. Nebe: G. Nebe and B. Venkov, “Unimodular lattices with long shadow. ” The paper is submitted to Journal of Number Theory. Full text can be found on the web page of G. Nebe. After Elkies and Gaulter, we study odd unimodular lattice $\Lambda $ in $\mathrm{R}^{n} $ , whose all characteristic vectors $\gamma $ have big norm $(\gamma, \gamma)\geq n16 $ , where $n=\dim\Lambda $. We say that such alattice is an $(n16)$lattice. By shadow theory, this condition is equivalent to the fact that the theta series of $\Lambda $ , which is a polynomial in two standard generators $\theta_{3} $ and $\triangle_{8} $ , is in fact asum of three monomials $\theta_{\Lambda}=\theta_{3}^{n}+A\theta_{3}^{n8}\triangle_{8}+B\theta_{3}\triangle_{8}^{2}$ with two constants $A $ and $B $. If $\Lambda $ has no elements of norm 1, then $A=2n $. If moreover $\min\Lambda\geq 3 $ , i.e. $\Lambda $ contains no roots, then $B $ is also fixed as afunction of $n $. That fixes $\theta_{\Lambda} $ and gives, for example, for the number of elements of norm 3and 4 $n_{3} = \frac{4}{3}n(n^{2}69n+1208) $, $n_{4}=2n(n^{3}94n^{2}+2783n 24425) $. Our main result is that such $(n16)$lattices without roots can exist only for $n\leq 46 $. (Previous bound by Gaulter [Gau] for $(n16)$lattices, possibly with roots, was $n\leq 290\overline{l} $). Our bound $n=46 $ is achieved, because the lattice $\Lambda_{0}=O_{23}\oplus O_{23} $ , where $O_{23} $ is the shorter Leech lattice, satisfies our conditions