A message comprises a set of symbols which are to be mapped onto binary code words for transmission. The code word set is {000000, 000111, 010010, 100001, 110011, 111000}. How many bit errors in a code word can be reliably detected? (Type a number only - no units.)

Q1. As discussed in the first assignment, it can be helpful to study the probability oferrors and therefore it is useful to know the cardinalities of sets of words and validcodewords — this is where combinatorics comes in.(a) In Question 3 of Assignment 1, it was stated that the set of all bit strings oflength four has 16 elements and the set of all bit strings of length 12 has4096. Justify these cardinalities.2 marks(b) Suppose that a code is made up six digits from the range 0–9 (inclusive)and satisfies the rules:• The first three digits must be odd (and non-zero).• No number is repeated in the first five digits.• The sixth digit is the remainder of the sum of the first five digits dividedby 10.i. How many valid codewords are there?4 marksii. What is the probability that a random selected six digit number is a validcodeword?2 marks(c) One common mistake humans make when reading out numbers is transpo-sitions. Transposition is the special case of swapping letters of a word givenby swapping adjacent letters. (In Assignment 1 we mentioned that ISBN-10could detect such errors.)How many distinguishable rearrangements (not just transpositions) of theISBN-10 code0-205-08005-7are there?2 marksTotal: 10 marksPage 3 of 8

In a communication system each data packet consists of 10001000 bits. Due to the noise, each bit may be received in error with probability 0.10.1. It is assumed bit errors occur independently. Find the probability that there are more than 120120 errors in a certain data packet.

For a (6, 3) systematic linear block code, the codeword comprises of m0,m1,m2 as the message bits and p0,p1,p2 as the parity bits. The coded bits are given as: c0=m0⊕m1, c1=m0⊕m2, c2=m1⊕m2. Which of the statement(s) is/are correct? The error-correcting capability of the code is 1. The error-detecting capability of the code is 2. The error-correcting capability of the code is 2. The error-detecting capability of the code is 3

Error correcting codes rely on the assumption that transmitted bits are more likely to arrive without error than with error(s). Under what circumstances might it be more likely for a received word to have errors than to be error-free?Question 1AnswerThis will not happen because the binomial distribution always peaks at 0It can occur with a high enough p(bit error), but such a communication channel would be unusableA short word and high p(error)A long word and high p(bit error)

If two codewords are a Hamming distanceIf two codewords are a Hamming distance dd apart, it willapart, it willrequirerequire dd singlesingle--bit errors to convert one into the other.bit errors to conve