5 research outputs found
On polynomials of small range sum
In order to reprove an old result of R\'edei's on the number of directions
determined by a set of cardinality in , Somlai proved that
the non-constant polynomials over the field whose range sums are
equal to are of degree at least . Here we characterise all
of these polynomials having degree exactly , if is large
enough. As a consequence, for the same set of primes we re-establish the
characterisation of sets with few determined directions due to Lov\'asz and
Schrijver using discrete Fourier analysis
A generalization of the cylinder conjecture for divisible codes
We extend the original cylinder conjecture on point sets in affine
three-dimensional space to the more general framework of divisible linear codes
over and their classification. Through a mix of linear
programming, combinatorial techniques and computer enumeration, we investigate
the structural properties of these codes. In this way, we can prove a reduction
theorem for a generalization of the cylinder conjecture, show some instances
where it does not hold and prove its validity for small values of . In
particular, we correct a flawed proof for the original cylinder conjecture for
and present the first proof for .Comment: 16 page