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Faculty of Information Engineering & Technology The Communications Department Course: Channel Coding [COMM 604] Dr. Tallal Elshabrawy Tutorial (3) Problem (1): A binary block code has code vectors in systematic form as given: 𝟎 𝟎 𝟏 𝒗= 𝟏 𝟏 𝟏 𝟎 [𝟎 𝟎 𝟏 𝟎 𝟏 𝟏 𝟎 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟎 𝟏 𝟎 𝟏 𝟎 𝟏 𝟎 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟎 𝟎 𝟎 𝟏 𝟏 𝟏 𝟏] a) What is the rate of this code? b) Write G & H of this code in systematic form? c) What is the min hamming distance of this code? d) How many errors can be detected and errors can be corrected? e) Compute the syndrome S of the received vector 𝒓 = [𝟏𝟎𝟏𝟎𝟏𝟏]. Problem (2): A linear block code is constructed by taking k information bits and generating a codeword of length 𝟐𝒌 + 𝟏 as follows: the first 𝒌 bits consist of the information bits, next 𝒌 bits repeat the information bits and the last bit is the xor of the first 𝒌 bits. a) Is this a systematic code? (Justify your answer) b) What is the number of valid codewords? c) Construct the G matrix of this code. d) Find the 𝒅𝒎𝒊𝒏 of this code. Faculty of Information Engineering & Technology The Communications Department Course: Channel Coding [COMM 604] Dr. Tallal Elshabrawy Problem (3): Consider a linear systematic (8, 4) channel code. a) Construct the parity check matrix for such code such that 𝒅𝒎𝒊𝒏 = 𝟒 b) What are the error correction and detection capabilities of this code? Problem (4): If 𝑨𝒋 , 𝒋 = 𝟎, 𝟏, 𝟐, … … … , 𝟕 defines the number of the codewords with hamming weight j of (7,4) Linear block code. Given: 𝑨𝟎 = 𝟏, 𝑨𝟏 = 𝑨𝟐 = 𝑨𝟓 = 𝑨𝟔 = 𝟎, 𝑨𝟒 = 𝟕, 𝑨𝟕 = 𝟏 a) Determine the 𝒅𝒎𝒊𝒏 of this code. b) Compute the probability of undetectable error patterns if the channel error probability is 0.05? Problem (5): A binary block code whose parity check matrix is: 𝟏 𝟎 𝑯= 𝟎 𝟎 [𝟎 𝟏 𝟏 𝟎 𝟎 𝟎 𝟏 𝟏 𝟏 𝟎 𝟎 𝟎 𝟏 𝟏 𝟏 𝟎 𝟏 𝟎 𝟏 𝟏 𝟏 𝟏 𝟏 𝟎 𝟏 𝟏 𝟎 𝟏 𝟏 𝟎 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟏 𝟎 𝟎 𝟏 𝟏 𝟎 𝟏 𝟎 𝟎 a) What is the min distance of this code? (Explain your answer) 𝟎 𝟏 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟎 𝟎 𝟏] Faculty of Information Engineering & Technology The Communications Department Course: Channel Coding [COMM 604] Dr. Tallal Elshabrawy b) Find a codeword of minimum weight. c) How many errors can be detected and corrected simultaneously? Problem (6): A systematic (6, 3) code whose generator matrix is: 𝟏 𝑮 = [𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏 𝟏 𝟎 𝟏 𝟏 𝟎 𝟏 𝟏] 𝟎 𝟏 a) Construct the standard array for this code and determine the correctable error patterns and their corresponding syndromes. b) Use the standard array to decode the following received words: 𝒓 = {𝟏𝟎𝟎𝟏𝟎𝟏, 𝟎𝟏𝟏𝟏𝟏𝟏, 𝟎𝟎𝟏𝟎𝟎𝟏} Problem (7): Consider a (7, k) code that could be used to correct all double and triple adjacent errors: a) What is the maximum rate for such code? b) Design the generator and parity check matrices for such code in part a) c) Construct the syndrome table for such code in part a)