81 research outputs found
The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6
The existence of an extremal code of length 72 is a long-standing open
problem. Let C be a putative extremal code of length 72 and suppose that C has
an automorphism g of order 6. We show that C, as an F_2-module, is the
direct sum of two modules, one easily determinable and the other one which has
a very restrictive structure. We use this fact to do an exhaustive search and
we do not find any code. This proves that the automorphism group of an extremal
code of length 72 does not contain elements of order 6.Comment: 15 pages, 0 figures. A revised version of the paper is published on
IEEE Transactions on Information Theor
Symmetries of weight enumerators and applications to Reed-Muller codes
Gleason's 1970 theorem on weight enumerators of self-dual codes has played a
crucial role for research in coding theory during the last four decades. Plenty
of generalizations have been proved but, to our knowledge, they are all based
on the symmetries given by MacWilliams' identities. This paper is intended to
be a first step towards a more general investigation of symmetries of weight
enumerators. We list the possible groups of symmetries, dealing both with the
finite and infinite case, we develop a new algorithm to compute the group of
symmetries of a given weight enumerator and apply these methods to the family
of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's
theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To
appear in Advances in Mathematics of Communication
Some new results on the self-dual [120,60,24] code
The existence of an extremal self-dual binary linear code of length 120 is a
long-standing open problem. We continue the investigation of its automorphism
group, proving that automorphisms of order 30 and 57 cannot occur. Supposing
the involutions acting fixed point freely, we show that also automorphisms of
order 8 cannot occur and the automorphism group is of order at most 120, with
further restrictions. Finally, we present some necessary conditions for the
existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application
Inhomogeneous minima of mixed signature lattices
We establish an explicit upper bound for the Euclidean minimum of a number
field which depends, in a precise manner, only on its discriminant and the
number of real and complex embeddings. Such bounds were shown to exist by
Davenport and Swinnerton-Dyer. In the case of totally real fields, an optimal
bound was conjectured by Minkowski and it is proved for fields of small degree.
In this note we develop methods of McMullen in the case of mixed signature in
order to get explicit bounds for the Euclidean minimum.Comment: To appear in the Journal of Number Theor
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