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

In [8], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterized. Using geometrical arguments, we characterize the codewords of small weight in the dual code of the code of points and generators of Q(5, q) and H(5, q). For the dual codes of the codes of Q(5, q), q even, and Q(4, q), q even, we investigate the codewords with the largest weights. We show that there exists an interval such that for every even number k in this interval, there is a codeword in the dual code of Q(5, q), q even, with weight k. For Q(4, q), q even, we show that there is an empty interval in the weight distribution of the dual of the code of Q(4, q). To prove this, we show that a blocking set of Q(4, q), q even, of size q + 1 + r, where 0 < r < (q+4)/6, contains an ovoid of Q(4, q), improving on [4, Theorem 9]. Finally, we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces ofQ(2n+1, q), Q(2n, q), Q−(2n+ 1, q), and H(n, q). For q even and k sufficiently small, we determine the maximum weight of these codes and characterize the codewords of maximum weight. We also link our results to sets of even type in these polar spaces.