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Every binary linear

[n, k, d]

code satisfies

n \geq \sum_{j = 0}^{k - 1} ⌈ \frac{d}{2 ^{j}} ⌉ .

Puncturing on the support of a minimum-weight codeword reduces the rank by one and leaves minimum distance at least

⌈ d /2 ⌉

, which proves the bound inductively.

Griesmer bound