Da… / M… / MATH3411-5249_00219 / Assessments Hub / Practice Tests 1-3 + BCH question (Chapters 1-6) Started on Friday, 22 November 2024, 10:34 AM State Finished Completed on Friday, 22 November 2024, 10:35 AM Time taken 11 secs Grade 0.00 out of 10.00 (0%) Question 1 Not answered Marked out of 1.00 Inspera: Safe Exam Browser Update Inspera Safe Exam Browser versions for Windows and Mac devices have been updated recently. Please ensure that you have the latest versions installed in your device before exams. For more information refer to the Current Student Page. × QUIZ Arithmetic coding is used with source symbols ,

and stop symbol

represented by probabilities ,

and

What can the message bab

be encoded as?

(Please provide a valid message (number).)   a b ∙ 0.6 0.2 0.2 ∙ The message can be any number in the interval

. A correct answer is , which can be typed in as follows: 0.6936 [0.6912, 0.696) 0.6936    Question 2 Not answered Marked out of 1.00 Question 3 Not answered Marked out of 1.00 Let

and .  You may assume that

is a eld. A BCH code is obtained from

by replacing a message

by the coecients of a polynomial

where and

is the minimal polynomial of , where

is a root of . Using this code, encode the message : Use the code to correct and decode the received message , assuming that at most two errors occurred in transmission: (x) = + x + 1 ∈ [x]m 1 x 4 Z 2 F = [x]/⟨ (x)⟩Z 2 m 1 F F ( , ,… , )c 8 c 9 c 14 C(x) ∈ [x]Z 2 C(x) (x)C I (x)C R m(x) = = = = (x) + (x)C R C I + +⋯+c 8 x 8 c 9 x 9 c 14 x 14 (x) mod m(x)C I (x) (x) = + + + + 1m 1 m 3 x 8 x 7 x 6 x 4 (x) = + + + x+ 1m 3 x 4 x 3 x 2 α 3 α (x)m 1 m = 0101100 y = 000010101000001 A correct answer is , which can be typed in as follows: 001010100101100 A correct answer is , which can be typed in as follows: 1001011 001010100101100 1001011 Calculate

:

Then, using Euler's Theorem or otherwise, calculate

:

  ϕ(361) mod 3612 361 A correct answer is , which can be typed in as follows: 342 A correct answer is , which can be typed in as follows: 116 342 116   Question 4 Not answered Marked out of 1.00 Question 5 Not answered Marked out of 1.00 Consider a radix

-error correcting (possibly non-linear) code

of length . What is the greatest possible value of , according to the Sphere-Packing Bound? 3 2 C n = 8 |C| Tip: The code

is not necessarily non-linear, so its size is less xed. Note that a code

with the parameters that we have here might not actually exist - but we are choosing to ignore this possibility for the purposes of this question. A correct answer is , which can be typed in as follows: 50 C C 50 Consider a binary channel with source symbols

and output symbols 

with

,       

,       

First nd , 

and  ; then use these to nd .

      [Round your solutions to 2 decimal places] { , }a 1 a 2 { , }b 1 b 2 P( ) =a 1 0.81 P( | ) =b 1 a 1 0.91 P( | ) =b 2 a 2 0.95 P( )a 2 P( )b 1 P( )b 2 H(B) P( ) =a 2 P( ) =b 1 P( ) =b 2 H(B) = A correct answer is , which can be typed in as follows: 0.19 A correct answer is , which can be typed in as follows: 0.75 A correct answer is , which can be typed in as follows: 0.25 A correct answer is , which can be typed in as follows: 0.82 0.19 0.75 0.25 0.82   Question 6 Not answered Marked out of 1.00 Question 7 Not answered Marked out of 1.00 A radix

instantaneous code (I-code) has codeword lengths (not necessarily in order) 1,2,3,3,

and Kraft-McMillan coecient

. What is the value of ?

4 ℓ K = 13 32 ℓ Hint: The Kraft-McMillan Theorem is very useful, and its 3-part proof is worth studying and understanding, even if it is probably the hardest of the (few slightly hard) proofs in the course. A correct answer is , which can be typed in as follows: 2 2 Consider a symmetric binary channel with constant bit-error probability , where errors in different positions are independent. Suppose that a codeword

is sent from the binary repetition code with codewords of length , and the word

is received. The probability that there is at least one error and that the error(s) in 

can be detected using a pure error detection strategy is: (No answer given)       p x 5 y y + 5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ pp 5 p 4 (1 − p) 2 p 3 (1 − p) 3 p 2 (1 − p) 4 10 ⋅ ⋅ + 5 ⋅ ⋅ p(1 − p) 3 p 2 (1 − p) 4 5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ pp 4 (1 − p) 2 p 3 (1 − p) 3 p 2 (1 − p) 4 10 ⋅ ⋅ + 5 ⋅ ⋅ p+(1 − p) 3 p 2 (1 − p) 4 (1 − p) 5 5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ p+p 4 (1 − p) 2 p 3 (1 − p) 3 p 2 (1 − p) 4 (1 − p) 5 Note: This question mostly just asks whether the code can detect given numbers of errors. A correct answer is: 5 ⋅ (1 − p) ⋅ + 10 ⋅ ⋅ + 10 ⋅ ⋅ + 5 ⋅ ⋅ pp 4 (1 − p) 2 p 3 (1 − p) 3 p 2 (1 − p) 4   Question 8 Not answered Marked out of 1.00 Question 9 Not answered Marked out of 1.00 A Markov source

has transition matrix   State the Huffman code Huff

for

associated to

and the 2nd column of : [Please write your answer as a list enclosed by square brackets; e.g., [00,11].]   S = { , , }s 1 s 2 s 3 M = ⎡ ⎣ ⎢ ⎢ 3 5 3 10 1 10 3 5 3 10 1 10 1 10 1 5 7 10 ⎤ ⎦ ⎥ ⎥ 2 S s 2 M Hint: Remember to sort the symbols rst! A correct answer is , which can be typed in as follows: [0,10,11] [0,10,11] Using Fermat factorisation, factor

into a product

where

. What is the value of

? n = 35581 n = ab 2 ≤ a < b b− a A correct answer is , which can be typed in as follows: 60 60   Question 10 Not answered Marked out of 1.00 Let

be the binary linear code with parity check matrix

and let

be a basis for . How many codewords does

contain?

C H = ⎡ ⎣ ⎢ ⎢ ⎢ 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 ⎤ ⎦ ⎥ ⎥ ⎥ B C B Hint: What is a basis? A correct answer is , which can be typed in as follows: 3 3 Previous Activity Jump to... Next Activity   51作业君版权所有