Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

Figure 1.  The application of Construction 3.5 to an example codeword $ c\in {\mathcal{C}}_1(2,q) $ of weight $ 9q-12 $. More specifically, we consider nine lines of $ {\rm{PG}}(2,q) $ and define the codeword $ c: = \alpha(a_0-a_1-a_2)+2\alpha\,\widetilde{a}+\beta(b_0-b_1-b_2-b_3)+3\beta\,\widetilde{b} $. For this specific example, we assume $ q $ not prime, $ q \geqslant529 $ if $ h = 2 $ (to be able to apply Result 1.3) and $ q \geqslant125 $, $ p\notin\{2,3,7,11,13\} $, if $ h>2 $. Furthermore, $ \alpha,\beta\in{\mathbb{F}}_q^* $ are two non-zero elements such that $ 3\alpha+4\beta = 0 $. Lines are clustered in four 'stages', each of which consists of 'clustering' the lines by following the rule of thumb described in Construction 3.5. Holes that are about to 'merge' clusters are indicated by squares instead of circles. In the first stage (top left), every line forms its own cluster. In the second stage (top right), the solid bold lines form one cluster, as well as the dashed bold lines; the remaining lines $ \widetilde{a} $ and $ \widetilde{b} $ form two clusters on their own. In the third stage (bottom left), the line $ \widetilde{a} $ gets merged into the solid bold cluster and the line $ \widetilde{b} $ gets merged into the dashed bold cluster. Finally, in the last stage (bottom right), both clusters get merged into one. To explain more clearly how this merging process works, consider the point $ A $ in the second stage. At this stage, $ A $ belongs to both the support of the solid bold line cluster $ \{a_0,a_1,a_2\} $ (with non-zero value $ -2\alpha $) and the support of the cluster $ \{\widetilde{a}\} $ (with non-zero value $ 2\alpha $), and thus meets Property $ 2. $ of Definition 3.4. Moreover, $ A $ is a hole of $ c $, as well as a hole of $ {{c}}_{|{{\mathcal{V}}}} $ for every other cluster (and therefore fulfils Property $ 1. $ and $ 3. $ of Definition 3.4). Hence, the clusters $ \{a_0,a_1,a_2\} $ and $ \{\widetilde{a}\} $ are adjacent and thus will get merged in the next stage as Construction 3.5 prescribes.

Figure 2.  An example of a codeword $ c\in {\mathcal{C}}_1(2,q) $, $ p>3 $, that proves the sharpness of the bound of Theorem 3.8 for the case $ n = 2 $, $ |{\mathbb{H}}_c^\infty| = 3 $ (see Theorem 3.10)