On the small weight codewords of the functional codes C_2(Q), Q a non-singular quadric

TL;DR

This paper characterizes small weight codewords in the functional code C_2(Q) associated with a non-singular quadric Q in projective space, linking them to intersections with specific singular quadrics and quantifying their count.

Contribution

It provides a precise description of small weight codewords in C_2(Q) and establishes their geometric origin related to singular quadrics formed by two hyperplanes.

Findings

Small weight codewords correspond to intersections with singular quadrics made of two hyperplanes.

The number of such small weight codewords is explicitly calculated.

The geometric characterization links code properties to quadric intersections.

Abstract

We study the small weight codewords of the functional code C_2(Q), with Q a non-singular quadric of PG(N,q). We prove that the small weight codewords correspond to the intersections of Q with the singular quadrics of PG(N,q) consisting of two hyperplanes. We also calculate the number of codewords having these small weights.