Small weight codewords of projective geometric codes II

Abstract

The $p$-ary linear code ${\mathcal{C}}_k(n,q)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\text{PG}(n,q)$. It is known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal{O}\left(q^{k-1}\right)$ exists that cannot be written as a linear combination of at most $\sqrt{q}$ rows of $A$. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if $q⩾32$ is a composite prime power, every codeword of ${\mathcal{C}}_k(n,q)$ up to weight $\mathcal{O}\left(q^k\sqrt{q}\right)$ is a linear combination of at most $\sqrt{q}$ rows of $A$. We also generalise this result to the codes ${\mathcal{C}}_{j,k}(n,q)$, which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces and $j$-spaces, $j<k$.

22 pages

References in corpus (2)

• A note on small weight codewords of projective geometric codes and on the smallest sets of even type
• Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

Cited by in corpus (1)

• Multisets with few special directions and small weight codewords in Desarguesian planes