e$)$ With a Hamming distance D of $8,$ how many bit errors can be detected? $($if no correction is

done$)?$ #detected $=$

$($formula$)=$

bits.

f$)$ With a Hamming distance D of $8,$ how many bit errors can be corrected?

#corrected $=$

$($formula$)=$

bits.

g$)$ In cyclic coding all bit patterns are expressed as polynoms in $GF(2).$ On the transmitter side

we can calculate the code word $c(x)$ by a multiplication of $c(x)=f(x{)}^{**}g(x),$ and on the re$-$

ceiver side we can get back the information word by $f(x)=c\frac{x}{g}(x).$ If this division does not

not work out even, i$.$e$.$ does have a non$-$zero remainder, then the code word was corrupted

with bit error$($s$)$ and the remainder itself constitutes the syndrom.

The generator polynom is here:

$g(x)=x^3+x+1$

$[10011$Symbols from a digital source shall be transmitted over a channel with bit errors. The possi$-$

ble symbols will be coded binary using a source coding scheme, then split into words of

length $m,$ then protected by a forward error correction scheme.

First we consider the source coding, see the diagram below. It uses a binary code tree for

mapping the symbols to bits. In this application case the symbols appear with different

probabilities $p_1(A),$ thus a variable but not fixed word length is used.

Suppose these $7$ symbols appear with the following probabilities:

a$)$ What is the entropy of the symbols $($best average bits$/$symbol$),$ assuming the probabilities

given here? Give your answer to four decimal digits $($e$.$g$.1\mathrm{.}0000).$

b$)$ Find a Huffman code for these symbols and write it into the last column of the table.

$($sketch the tree on a separate sheet of paper, last page$)$

c$)$ Find the average code length assuming the Huffman code is used to encode these sym$-$

bols: $L=$

d$)$ Suppose that $6$ symbols are enco$-$

ded using the right tree $($not the

same as above$),$ find the symbol se$-$

quence corresponding to this bit

stream:

$0011011010000111010011$

Please input the corresponding characters

below: $($symbol sequence corresponding to

this bit stream$)$: