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Weights of Codewords of a Binary Linear Code

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The weight of w(x) of a vector x in (F_q )^n is defined to be the number of nonzero entries of x.

Prove that, in a binary linear code, either all the codewords have even weight or exactly half even weight and half odd weight.

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We prove that either all codewords of a binary linear code have even weight or that half have even weight and the other half have odd weight.

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We wish to prove that in a binary linear code, either all codewords have even weight or half have even weight and half have odd weight.
Let C be a binary linear code. Consider the function defined by

Note that ...

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