Displaying similar documents to “Codes, lattices, and Steiner systems.”

Universal codes and unimodular lattices

Robin Chapman, Patrick Solé (1996)

Journal de théorie des nombres de Bordeaux

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Binary quadratic residue codes of length p + 1 produce via construction B and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction A modulo 4 . We prove in a direct way the equivalence of these two constructions for p ≤ 31 . In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice B W 32 . The equivalence between construction B modulo 4 plus density doubling...

Another 80-dimensional extremal lattice

Mark Watkins (2012)

Journal de Théorie des Nombres de Bordeaux

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We show that the unimodular lattice associated to the rank 20 quaternionic matrix group SL 2 ( F 41 ) ⊗ S ˜ 3 ⊂ GL 80 ( Z ) is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the Θ -series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner...

On extremal additive 𝔽 4 codes of length 10 to 18

Christine Bachoc, Philippe Gaborit (2000)

Journal de théorie des nombres de Bordeaux

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In this paper we consider the extremal even self-dual 𝔽 4 -additive codes. We give a complete classification for length 10 . Under the hypothesis that at least two minimal words have the same support, we classify the codes of length 14 and we show that in length 18 such a code is equivalent to the unique 𝔽 4 -hermitian code with parameters [18,9,8]. We construct with the help of them some extremal 3 -modular lattices.

Lattice-Like Total Perfect Codes

Carlos Araujo, Italo Dejter (2014)

Discussiones Mathematicae Graph Theory

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A contribution is made to the classification of lattice-like total perfect codes in integer lattices Λn via pairs (G, Φ) formed by abelian groups G and homomorphisms Φ: Zn → G. A conjecture is posed that the cited contribution covers all possible cases. A related conjecture on the unfinished work on open problems on lattice-like perfect dominating sets in Λn with induced components that are parallel paths of length > 1 is posed as well.

Lattice of ℤ-module

Yuichi Futa, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding...