COD Section $5\mathrm{.}3$ shows the typical method to index a direct$-$mapped cache, specifically $($Block address$)$ modulo $($Number of blocks in the cache$).$ Assuming a $64-$bit address and $1024$ blocks in the cache, consider a different indexing function, specifically $($Block address$[63$:$54]$ XOR Block address$[53$:$44]).$ Is it possible to use this to index a direct$-$mapped cache? If so$,$ explain why and discuss any changes that might need to be made to the cache. If it is not possible, explain why.

$($a$)$

Calculate the total number of bits required to implement a $32$ KiB cache with two$-$word blocks.

$($b$)$

Calculate the total number of bits required to implement a $64$ KiB cache with $16-$word blocks. How much bigger is this cache than the $32$ KiB cache described in Part a$?($Notice that, by changing the block size, we doubled the amount of data without doubling the total size of the cache.$)$

$($c$)$

Explain why this $64$ KiB cache, despite its larger data size, might provide slower performance than the first cache.

$($d$)$

Generate a series of read requests that have a lower miss rate on a $32$ KiB two$-$way set associative cache than on the cache described in Part a$.$