## Parameters of a specific code

### Task number: 4033

Consider a binary code containing all words of length \( n \) of even weight, that is \( C = \{x \in \mathbb Z_2^n: w (x) \equiv 0 \mod 2 \} \). Determine the parameters of the code \( C \) and describe its orthogonal complement \( C^{\bot} \).

#### Answer

It is \( (n, n-1{,}2) \)-code, the complement has dimension \( 1 \) and is generated by the vector \( (1{,}1, \ldots, 1) \).