A $[31,22,5]-$linear binary code is supposed not to exist and the proof can be given by contradiction. By determining the number of words of weight $3$ in:
1. cosets of coset leader with weight $2$
2. cosets of coset leader of weight $3$, one should arrive at a number smaller than $\binom{31}{3}$ (total number of weight $3$ words).
In the first case, the only case of having a word of weight $3$ in the coset would be by combining the coset-leader with a word of weight $5$, but bounding the number of words of weigth $5$ is proving to be difficult and I was told by my professor I wouldn't even need to do it. It should be straightforward even assuming that the code has no problems with conditions on the number of words of weight $5$.
Can anyone provide help on the boundings?
2026-03-25 11:07:59.1774436879