The C++ code
The program is based on Proposition 5.5 which asserts that a maximal sequence contains $(e_0 e_1 e_2 e_3 g)^4$. We denote these five vectors by col...col. Each of them can be choosen no more than 4 times. The rest vectors are denoted by col...col, each of which can be choosen only once.
The idea is to search all posibilities of col...col to find an admisible sequence of length 22. Here, admisible means that each summation of 5 vectors is not zero vector. If there is no such sequence, then we prove the conjecture.
The program also output all admisible sequence of length 21. Furthermore, we check whether or not it is a sequence of the form $(e_0 e_1 ... e_8)^4. If not, we call it a special sequence. From the output file we see that there is no special sequences.The output file