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On the weight distribution of binary linear codes (Corresp.)

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Let V be a binary linear (n,k) -code defined by a check matrix H with columns h_{1}, \cdots ,h_{n} , and let h(x) = 1 if x \in \{h_{1}, \cdots , h_{n}\} , and h(x) = 0 if x \in \neq {h_{1}, \cdots ,h_{n}} . A combinatorial argument relates the Walsh transform of h(x) with the weight distribution A(i) of the code V for small i(i< 7) . This leads to another proof of the Pless i th power moment identities for i < 7 . This relation also provides a simple method for computing the weight distribution A(i) for small i . The implementation of this method requires at most (n-k+ 1)2^{n-k} additions and subtractions, 5 . 2^{n-k} multiplications, and 2^{n-k} memory cells. The method may be very effective if there is an analytic expression for the characteristic Boolean function h(x) . This situation will be illustrated by several examples.

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Information Theory, IEEE Transactions on  (Volume:25 ,  Issue: 1 )