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

In [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr. 42 (1) (2007) 73–92], 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 , q 2 ) , 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 q 2 + 1 + r , where 0 < r < ( q + 4 ) / 6 , contains an ovoid of Q ( 4 , q ) , improving on [J. Eisfeld, L. Storme, T. Szőnyi, P. Sziklai, Covers and blocking sets of classical generalised quadrangles, Discrete Math. 238 (2001) 35–51, Theorem 9].