经典广义四边形与经典极空间对偶码中的码字
On codewords in the dual code of classical generalised quadrangles and classical polar spaces
摘要：
在[9]中，刻画了Q（4，q）的点、线码的对偶码中的小权码字。受这一结果的启发，利用几何参数，我们刻画了点码与Q+（5，q）和H（5，q2）的生成元的对偶码中的小权重码字，并给出了经典极空间的点码与k-空间的对偶码中码字权重的下界。此外，我们调查的码字与最大的重量在这些代码中，其中q甚至和k足够小，我们确定的最大重量和最大重量的码字。证明了存在一个区间，使得对于该区间中的每个偶数w，Q+（5，q），q偶数的对偶码中存在一个码字，其权重为w，并且证明了Q（4，q），q偶数的对偶码的权重分布中存在一个空区间.为了证明这一点，我们证明了Q（4，q）的一个大小为q2 + 1 +r（其中0 < r <（q +4）/6）的阻塞集（q为偶数）包含Q（4，q）的一个卵形体，改进了[5，定理9].
英文摘要：
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].
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