Minimal codewords arising from the incidence of points and hyperplanes in projective spaces
Daniele Bartoli, Lins Denaux · Advances in Mathematics of Communications · 2021
Over the past few years, the codes $ {\mathcal{C}}_{n-1}(n,q) $ arising from the incidence of points and hyperplanes in the projective space $ {\rm{PG}}(n,q) $ attracted a lot of attention. In particular, small weight codewords of $ {\mathcal{C}}_{n-1}(n,q) $ are a topic of investigation. The main result of this work states that, if $ q $ is large enough and not prime, a codeword having weight smaller than roughly $ \frac{1}{2^{n-2}}q^{n-1}\sqrt{q} $ can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.