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2 -modular lattices from ternary codes

Robin Chapman, Steven T. Dougherty, Philippe Gaborit, Patrick Solé (2002)

Journal de théorie des nombres de Bordeaux

The alphabet 𝐅 3 + v 𝐅 3 where v 2 = 1 is viewed here as a quotient of the ring of integers of 𝐐 ( - 2 ) by the ideal (3). Self-dual 𝐅 3 + v 𝐅 3 codes for the hermitian scalar product give 2 -modular lattices by construction A K . There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual...

A Necessary and Sufficient Condition for the Existence of an (n,r)-arc in PG(2,q) and Its Applications

Hamada, Noboru, Maruta, Tatsuya, Oya, Yusuke (2012)

Serdica Journal of Computing

ACM Computing Classification System (1998): E.4.Let q be a prime or a prime power ≥ 3. The purpose of this paper is to give a necessary and sufficient condition for the existence of an (n, r)-arc in PG(2, q ) for given integers n, r and q using the geometric structure of points and lines in PG(2, q ) for n > r ≥ 3. Using the geometric method and a computer, it is shown that there exists no (34, 3) arc in PG(2, 17), equivalently, there exists no [34, 3, 31] 17 code.This research was partially...

A new lower bound for the football pool problem for 7 matches

Laurent Habsieger (1996)

Journal de théorie des nombres de Bordeaux

Let K 3 ( 7 , 1 ) denote the minimum cardinality of a ternary code of length 7 and covering radius one. In a previous paper, we improved on the lower bound K 3 ( 7 , 1 ) 147 by showing that K 3 ( 7 , 1 ) 150 . In this note, we prove that K 3 ( 7 , 1 ) 153 .

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