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'''''High-level proof:'''''
To show the existence of the linear code that satisfies those constraints, the [[probabilistic method]] is used to construct the random linear code. Specifically the linear code is chosen randomly by choosing the [[random generator|random]] generator matrix <math>G</math> in which the element is chosen uniformly over the field <math>\mathbb{F}_q^n </math>. Also the [[Hamming distance]] of the linear code is equal to the minimum weight of the [[Code word (communication)|codeword]]. So to prove that the linear code generated by <math>G</math> has Hamming distance <math>d</math>, we will show that for any <math>m \in \mathbb{F}_q^k \smallsetminus \left\{ 0 \right\}, \operatorname{wt}(mG) \ge d</math> . To prove that, we prove the opposite one; that is, the probability that the linear code generated by <math>G</math> has the Hamming distance less than <math>d</math> is exponentially small in <math>n</math>. Then by probabilistic method, there exists the linear code satisfying the theorem.
'''''Formal proof:'''''
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