Question $5$: Some language has $256$ letters. We are also given some text in this

language. Let num$($x$)$ be the number of times a letter x appears in the text. Assume

that for every x and y$,$ both num$($y$)$ num$($x$)/2$ and num$($x$)$ num$($y$)/2$ hold. In

the $8-$bit representation, every letter is given an $8-$bit value. Note that $28=256,$ and

thus we can represent $256$ letters with $8$ bits. Is the Huffman code always better in

this case? Note that in the $8-$bit representation, every letter is given the same length.

The question is whether using different length codes, as in Huffman coding, is always

an improvement.

$2$