 MacWilliams identity

ID: macwilliams-identity

k

and its dual code, the weight enumerator convention

s^{wt (x)} t^{n - wt (x)}

gives

W_{C^{⊥}} (s, t) = 2^{- k} W_{C} (t - s, t + s) .

To prove it, insert

1_{C^{⊥}} (y) = 2^{- k} \sum_{x \in C} (- 1)^{x \cdot y}

into the enumerator sum. Summing independently over each coordinate of

y

yields a factor

t + s

x_{j} = 0

and

t - s

x_{j} = 1

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