Pair codes

A friend of yours is quite excited about having developed a new class of codes: pair codes. Pair codes are a generalization of the $2-$out$-$of$-5$ representation discussed in Notes Set $4\mathrm{.}2($Lumetta$\text{'}$s notes$).$ In a pair code, each code word has exactly two $1$ bits. However, one can define a pair code on any number of bits $N.$ For example, if $N=100,$ one has $100$ bits in the code words, and exactly two $1$ bits.

Your friend points out that as $N$ grows, the fraction of valid code words drops dramatically. For $N=100,$ for example, there are only $4950$ valid code words $(100$ times $99$ divided by $2),$ but there are $2100$ bit patterns. Your friend argues that error correction capabilities for pair codes must be quite powerful, since the codes are so sparse

What is the Hamming distance of the pair code with $6-$bit code words? Use an example to prove that your answer is correct.

What is the Hamming distance of the pair code on $100-$bit code words? Explain how you can again prove that your answer is correct $($please avoid writing $100-$bit numbers$).$

How many bits can be corrected using a pair code with $N-$bit code words?