Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$ is denoted by $m(k,q)$. Here we determine $m(7,2)$, $m(8,2)$, and $m(9,2)$, as well as full classifications of all codes attaining $m(k,2)$ for $k\le 7$ and those attaining $m(9,2)$. For $m(11,2)$ and $m(12,2)$ we give improved upper bounds. It turns out that in many cases attaining extremal codes have the property that the weights of all codewords are divisible by some constant $\Delta>1$. So, here we study the minimum lengths of minimal codes where we additionally assume that the weights of the codewords are divisible by $\Delta$.

翻译：极小码是一类所有非零码字均为极小码字的线性码，即其支撑集不包含于另一码字的支撑集中。此类$k$维线性码在域$\mathbb{F}_q$上的最小可能长度记为$m(k,q)$。本文确定了$m(7,2)$、$m(8,2)$和$m(9,2)$的值，同时完成了对满足$k\le 7$时所有达到$m(k,2)$的码的完全分类，以及对达到$m(9,2)$的码的完全分类。针对$m(11,2)$和$m(12,2)$，我们给出了改进的上界。结果表明，在多数情形下，达到极值的码具有如下性质：所有码字的权重均可被某个常数$\Delta>1$整除。因此，本文进一步研究了在额外假设码字权重可被$\Delta$整除的条件下极小码的最小长度。