Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

Abstract

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 $\text{PG}(n,q)$ 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.

[v1] 18 pages [v2] 19 pages, 2 figures: added some clarifications and practical examples to the second part of the paper

Cited by in corpus (2)

• A note on small weight codewords of projective geometric codes and on the smallest sets of even type
• Small weight codewords of projective geometric codes II