Problem 8 - Coding Theory

Consider an $n$-dimensional vector space over the finite field $\mathbb{Z}_2$ of size two. Define the distance between two vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ in this vector space as the number of coordinates that have different values.

1.

Let $n=5$. Construct a vector subspace consisting of four vectors in which any pair of distinct vectors has distance at least three.

2.

Let $n=15$. Construct a vector subspace consisting of 32 vectors in which any pair of distinct vectors has distance at least five.