Let $C$ be a binary linear code. Explain what it means for $C$ to have length $n$ and $\operatorname{rank} k$. Explain what it means for a codeword of $C$ to have weight $j$.

Suppose $C$ has length $n$, rank $k$, and $A_{j}$ codewords of weight $j$. The weight enumerator polynomial of $C$ is given by

$$W_{C}(s, t)=\sum_{j=0}^{n} A_{j} s^{j} t^{n-j}$$

What is $W_{C}(1,1) ?$ Prove that $W_{C}(s, t)=W_{C}(t, s)$ if and only if $W_{C}(1,0)=1$.

Define the dual code $C^{\perp}$ of $C$.

(i) Let $\mathbf{y} \in \mathbb{F}_{2}^{n}$. Show that

$$\sum_{\mathbf{x} \in C}(-1)^{\mathbf{x} \cdot \mathbf{y}}= \begin{cases}2^{k}, & \text { if } \mathbf{y} \in C^{\perp} \\ 0, & \text { otherwise }\end{cases}$$

(ii) Extend the definition of weight to give a weight $w(\mathbf{y})$ for $\mathbf{y} \in \mathbb{F}_{2}^{n}$. Suppose that for $t$ real and all $\mathbf{x} \in C$

$$\sum_{\mathbf{y} \in \mathbb{F}_{2}^{n}} t^{w(\mathbf{y})}(-1)^{\mathbf{x} \cdot \mathbf{y}}=(1-t)^{w(\mathbf{x})}(1+t)^{n-w(\mathbf{x})}$$

For $s$ real, by evaluating

$$\sum_{\mathbf{x} \in C}\left(\sum_{\mathbf{y} \in \mathbb{F}_{2}^{n}}(-1)^{\mathbf{x} \cdot \mathbf{y}}\left(\frac{s}{t}\right)^{w(\mathbf{y})}\right)$$

in two different ways, show that

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