Exercise-3 - ( X g , using modulo 2 arithmetic. 4) Show...

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THE UNIVERSITY OF WESTERN ONTARIO FACULTY OF ENGINEERING ECE-433b-COMMUNICATION SYSTEMS Exercise-3 [Release Date: January 30, 2007] Topics Covered ( January 23, 24, and 26, 2007 Syndrome Decoding and Standard Array for decoding Linear Block Codes. Examples of Error Correction Capability of Block Codes. Definition of Cyclic Codes, Cyclic Redundancy Check (CRC) Codes & their error detection capability. Capability of BCH codes 1) Consider the CRC-16 code described by the generator polynomial 1 2 15 16 + + + X X X . Using examples demonstrate the various error detection properties of the code. 2) Consider the (5, 1) repetition code. Evaluate the syndrome s for the following error patterns: a. All five possible single-error patterns; b. All 10 possible double-error patterns. 3) Let 1 ) ( 2 4 + + + = X X X X g and let 1 ) ( 3 + + = X X X m . Find the remainder when ) ( 4 X m X is divided by )
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Unformatted text preview: ( X g , using modulo 2 arithmetic. 4) Show that if ) ( X g contains the factor X + 1 , then all error sequences with an odd number of errors are detected. 5) Consider a (6, 2) code generated by the matrix: 1 1 1 1 1 1 1 1 a. Construct the code table for this code and determine the minimum distance between the code words. b. Prepare a suitable decoding table. This code can correct all single-error patterns, seven double-error patterns, and two triple-error patterns. 6) A CRC is constructed to generate 4 check bits for an 11-bit message given by: 1 0 1 0 1 1 1 0 0 1 1. The generator polynomial is given by: 1 3 4 + + X X . Compute the check bits. 7) Encode the message 101 using polynomial division and the generator 4 2 1 ) ( X X X X g + + + = Try these problems yourself; no need to submit solutions...
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