On codewords in the dual code of classical generalised quadrangles and classical polar spaces

TL;DR

This paper characterizes small and large weight codewords in the dual codes of classical polar spaces, providing bounds, explicit constructions, and insights into their weight distributions using geometric methods.

Contribution

It extends previous work by characterizing small weight codewords in dual codes of higher-dimensional polar spaces and analyzing their weight distribution and maximum weight codewords.

Findings

Characterization of small weight codewords in dual codes of Q+(5, q) and H(5, q2)

Lower bounds on weights of codewords in duals of point and k-space codes

Existence of intervals of weights with corresponding codewords in dual codes

Abstract

In [9], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q+(5, q) and H(5, q2), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q+(5, q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4, q), q even. To prove this, we show that a blocking set of Q(4, q), q even, of size q2 +1+r, where 0 < r < (q +4)/6, contains an ovoid of Q(4, q), improving on [5, Theorem 9].