53 research outputs found
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
Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface
In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on
the intersection of a surface of degree and a non-degenerate Hermitian
surface in \PP^3(\Fqt). The conjecture was proven to be true by Edoukou in
the case when . In this paper, we prove that the conjecture is true for
and . We further determine the second highest number of rational
points on the intersection of a cubic surface and a non-degenerate Hermitian
surface. Finally, we classify all the cubic surfaces that admit the highest and
second highest number of points in common with a non-degenerate Hermitian
surface. This classifications disproves one of the conjectures proposed by
Edoukou, Ling and Xing
Maximum number of -rational points on nonsingular threefolds in
We determine the maximum number of -rational points that a
nonsingular threefold of degree in a projective space of dimension
defined over may contain. This settles a conjecture by Homma and
Kim concerning the maximum number of points on a hypersurface in a projective
space of even dimension in this particular case.Comment: 8 page
- …