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

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.