$\left(U+V\mid V\right)$ codes

We will be interested here in binary $\left(U+V\mid V\right)$ codes. A binary $\left(U+V\mid V\right)$ construction takes two binary codes of a same length $n$ and produces a code of length $2n$ as follows (we denote here the concatenation of two vectors $\mathbf{x}$ and $\mathbf{y}$ by $(\mathbf{x}\vert\mathbf{y})$)

Definition 1 ( $\left(U+V\mid V\right)$ binary code)   Let $U$ and $V$ be two binary linear codes of a same length. We define the $\left(U+V\mid V\right)$-construction of $U$ and $V$ as the binary linear code:

$$\left(U+V\mid V\right)= \left\{ (\mathbf{u}+ \mathbf{v}\mid \mathbf{v}); \mathbf u \in U \hbox{ and } \mathbf v \in V\right\}.$$

The dimension of the $\left(U+V\mid V\right)$ code is $k_U+k_V$ and its minimum distance is $\min(2d_V,d_U)$ when the dimensions of $U$ and $V$ are $k_U$ and $k_V$ respectively, the minimum distance of $U$ is $d_U$ and the minimum distance of $V$ is $d_V$.

A polar code is an iterated $\left(U+V\mid V\right)$ code, in the sense that the codes $U$ and $V$ are themselves $\left(U+V\mid V\right)$ codes and so and so forth up to the point where the constituent codes are of length $1$. Such codes have therefore a length which is a power of $2$. For instance a polar code of length $4$ is a $\left(U+V\mid V\right)$ code of length $4$ where $U$ is a code of length $2$ which is a $\left(U+V\mid V\right)$ construction obtained from two codes of length $1$. The same applies to the code $V$ of length $2$. A polar code of length $2^n$ and dimension $k$ is associated in a natural way to a binary tree of depth $n$ with $k$ leaves corresponding to the information bits of the code. It is first asked to show the two properties of a $\left(U+V\mid V\right)$ code (about their dimension and their minimum distance). Use the property about the minimum distance and the dimension to find a polar code of length $8$, dimension $4$ and minimum distance $4$.