Is it true that in a binary linear code number of words, which start with zero, is always at least as much as words, which start with 1?

I have no idea how to prove or disprove it. Can you give me a hint how to start, please?

  • $\begingroup$ Induction on the dimension of the code. (Base case: the code of dimension 0, which contains only the all-0 word.) $\endgroup$ – fkraiem May 5 at 14:40
  • $\begingroup$ And of course any code of dimension $n \ge 1$ can be written as the direct sum of two codes, of dimensions $n-1$ and $1$. $\endgroup$ – fkraiem May 5 at 14:48

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