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Linearity guarantees that the minimum [[Hamming distance]] ''d'' between a codeword ''c''<sub>0</sub> and any of the other codewords ''c'' ≠ ''c''<sub>0</sub> is independent of ''c''<sub>0</sub>. This follows from the property that the difference ''c'' − ''c''<sub>0</sub> of two codewords in ''C'' is also a codeword (i.e., an [[element (mathematics)|element]] of the subspace ''C''), and the property that ''d''(''c'', c<sub>0</sub>) = ''d''(''c'' − ''c''<sub>0</sub>, 0). These properties imply that
:<math>\min_{c \in C,\ c \neq c_0}d(c,c_0)=\min_{c \in C,\ c \neq c_0}d(c-c_0, 0)=\min_{c \in C,\ c \neq 0}d(c, 0)=d.</math>
In other words, in order to find out the minimum distance between the codewords of a linear code, one would only need to look at the non-zero codewords. The non-zero codeword with the smallest weight has then the minimum distance to the zero codeword, and hence determines the minimum distance of the code.
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