18 Henry Quastler 



binary representation per event and per symbol. Then, the general condition 

 of representability can be stated as follows : 



^y ' f^u ^ ^X ' ^X 



EXERCISES 



1. A weakness of the Paul Revere code is that there is no positive signal for "peace and 

 quiet". Hence, the colonists could not be sure whether the absence of a warning signal meant 

 "peace and quiet" or a disturbance in the communication system. Show how two lights 

 could be used to indicate the four situations by positive signals. 



2. Any integer can be written as a sum of powers of 2(1,2,4,8,16, • • •)• For instance: 



27 = 16 + 8 + 2 + 1 

 = 2* + 2=' + 21 + 2» 



In binary notation, one indicates the power by position, and writes a '1' in appropriate position 

 if this power does enter the sum, a '0' if it does not. Thus, '27' becomes 11011. 



(a) Write the following numbers in binary notation: 0,1,2,3,4,5,6,7,8,9,10,12,16,1955. 



(b) Write the following binary numbers in decimal notation: 1001, 1011, 10010011, 100000. 

 Any proper fraction can be written as a sum of powers of 1/2, (1/2, (1/2)^ = 1/4, (1/2)^ = 1/8, 

 etc.). For instance: .75 = 1/2 + 1/4, or, in binary notation, .11. 



(c) Translate into decimal notation: .001, .1001001 



3. (a) encode the message 'ABCDE' in code (a) of the five-word codes described earlier, 

 (b) decode the message: '000001011011' in code (b). 



4. This assignment is coded in the Fano code for English letters given earlier. 

 001001001 10000101 1 1001 1 10001 10001001 10101001001001 1000001010001 1001010 

 1 101001 10000010101 1111111 1001001001001 1 1 1 1 10001 10010101 101000000101001 

 0101 100000001 1 1 100000101 10100010101 1 1000100001 1 101 1000010101 101 1001010 

 0101 100101 1 1 1 1 1 101001000100001 101111 101 101 110111 101 1000001 1 10010010010 

 001 1 100001 110101111111 1001001 1 100001 1111011 100101 100101 1 10001 1 1 101 1000 

 001001 1 1 100100101 1 10010001 100101001001001001 1 1 1 1 100001001 101 1001001 100 

 100101 1 10100100101 1 1001001 1 1 10 100000 1 10010000001 1 1 10000010001001 1 1 1000 

 001001001001 1 101 101000000101 101 1000001 1 1001 1 1 1 1 1001001001001 1 101 101000 

 000 101 1010001 1 1 1 1 101010101 101000101 1 1 1001 1001 1000000001 1 1 1 1 10010001 10 

 010110011000 111 



(This assignment is very tedious but it is good practice.) 



5. Given a real situation with three categories and probabilities p(A) = .8, p{B) = .15, 

 p(C) = .05. Construct a binary code which comes within 10 per cent of the minimum bulk. 



6. A protein is thought to be a linear arrangement of amino acids of which there are 

 (about) twenty kinds in each cell. The specificity of a protein depends mostly on the sequence 

 of amino acids, i.e. a protein can be considered as a 'message' written in a twenty-letter 

 alphabet. It is known that, in the living cell, protein specificity is determined by nucleic 

 acids. These are linear arrangements of nucleotides, of which there are four different kinds. 



Question: what is the minimum number of nucleotides needed, on average, to specify 

 each amino acid? Assume all amino acids to be equiprobable. 



III. THE MEASURE OF INFORMATION OR UNCERTAINTY 



It seems reasonable to equate the amount of information acquired, as a 

 result of an event, to the amount of uncertainty which its occurrence has 



