Consider a (7, 4) code whose generator matrix is
Consider a (7, 4) code whose generator matrix is
a) Find all the codewords of the code
b) Find H, the parity check matrix of the code.
c) Compute the syndrome for the received vector 1 101 1 0 1. Is this a valid code vector?
d) What is the error-correcting capability of the code?
e) What is the error-detecting capability of the code?
Homework Answers
Let G- be a generator matrix for a block code (not necessarily a "good" code) a) b) c) What is th...
Let G- be a generator matrix for a block code (not necessarily a "good" code) a) b) c) What is the n, k, the rate and the bandwidth expansion for this code? Find the parity check matrix H )Build the standard array for the code. Assume the coset leaders are vectors with one "l", starting from the left side of the vector, i.e., the first coset leader will be (1 0...), the second (01 0 ...) starting again from the...
Consider the (5,2) linear binary code, C, with linear space of codewords spanned by the codewords...
Consider the (5,2) linear binary code, C, with linear space of codewords spanned by the codewords (1, 0, 1,1, 1) and (0, 1, 1, 1, 0). 4. Find all codewords in C, find the systematic generator matrix, G, and a parity check matrix, H, for the code. a. Determine dmin for the code and the code's weight distribution. Determine all codewords in the dual code, Cd . Find a systematic generator matrix, Ga, for the dual code, and corresponding parity...
Design (7,3) linear block code with parity check matrix given as H = 0 1 11 0 0 1 1 0 10 1 0 1 1 1 00 0 1 1 a. Find all...
Design (7,3) linear block code with parity check matrix given as H = 0 1 11 0 0 1 1 0 10 1 0 1 1 1 00 0 1 1 a. Find all the corresponding codewords of the code. b. What is the error the error-correcting and error-detection capabilities of the code? c. Find the syndrome for the received vector R = [1101011]. d. Assuming the receiver Maximum likelihood algorithm construct syndrome table for the correctable error patterns
Consider the 2-error correcting, narrow-sense RS code over GF(16) (α is a primitive element). (a) Write down the generator polynomial and the parity check polynomial. (b) Provide a parity check matrix...
Consider the 2-error correcting, narrow-sense RS code over GF(16) (α is a primitive element). (a) Write down the generator polynomial and the parity check polynomial. (b) Provide a parity check matrix for the code. (c) Decode the received vector V = [α6 α12 α9 α12 0 0 0 α8 0 0 0 α10 α α13 α].
1. (30 points) Consider the systematic binary linear (6,3) code with generator matrix 1 0 01...
1. (30 points) Consider the systematic binary linear (6,3) code with generator matrix 1 0 01 1 0 G- 0 1 0 0 1 1 a) Determine the parity check matrix H of the code. b) What is the minimum distance of the code? How many errors can this code correct and detect? c) Show the results in b) using decoding table d) Find the most likely codeword, given that the noisy received codeword is 010101. e) Now suppose 001101...
(c) Consider the (7, 4) Hamming Code defined by the generator polynomial g(x)-1 +x+x'. The code...
(c) Consider the (7, 4) Hamming Code defined by the generator polynomial g(x)-1 +x+x'. The code word 1000101 is sent over a noisy channel, producing the received word 0000101 that has a single error. Determine the syndrome polynomial s(x) for this received word. Find its corresponding message vector m and express m in polynomial m(x). 0
Consider a generator matrix G for a nonsystematic (6, 3) code:
Consider a generator matrix G for a nonsystematic (6, 3) code:Construct the code for this G, and show that dmin, the minimum distance between codewords is 3, Consequently, this code can correct at least one error.
coding theory 1. If 100 031 Go 01 0 0 9 0001 27 01 0 054 is a generator matrix for a linear code over Fi encode the message stream m(2, 3,9, 6, 1,4,3, 8) (by breaking it into encodable pieces). 2....
coding theory
1. If 100 031 Go 01 0 0 9 0001 27 01 0 054 is a generator matrix for a linear code over Fi encode the message stream m(2, 3,9, 6, 1,4,3, 8) (by breaking it into encodable pieces). 2. If T (1 0 1 2 3 4 be the transpose of a parity-check matrix for a perfect 1-error-correcting code over Fs, with implicit generator matrix 4410 0 0 3 4 0 1 0 0 1400 01
1....
Consider the (4,1) repetition code
Consider the (4,1) repetition code. (a) show that HG^T=0 (b) Compute the syndromes for all one error patterns (c) What are the two transmitted codewords for a two-bit message m = 0 1 (d) If the received codewords is 0010 what is the decoded message bit using syndrome decoding
1. Channel Coding We would like to add linear block code (3,6) using the generator matrix:...
1. Channel Coding We would like to add linear block code (3,6) using the generator matrix: 1 001 01 G-0 1 0 0 1 1 (a) (5 points) Determine the parity check matrix H (b) (20 points) What is the minimum distance of this code? How many error can this code correct? (c) (5 points) What is the code word for the data sequence 011000101111? (d) (20 points) If you receive the codeword 010001000010101010, what is the transmitted sequence?
Let G- be a generator matrix for a block code (not necessarily a "good" code) a) b) c) What is the n, k, the rate and the bandwidth expansion for this code? Find the parity check matrix H )Build the standard array for the code. Assume the coset leaders are vectors with one "l", starting from the left side of the vector, i.e., the first coset leader will be (1 0...), the second (01 0 ...) starting again from the...
Consider the (5,2) linear binary code, C, with linear space of codewords spanned by the codewords (1, 0, 1,1, 1) and (0, 1, 1, 1, 0). 4. Find all codewords in C, find the systematic generator matrix, G, and a parity check matrix, H, for the code. a. Determine dmin for the code and the code's weight distribution. Determine all codewords in the dual code, Cd . Find a systematic generator matrix, Ga, for the dual code, and corresponding parity...
1. (30 points) Consider the systematic binary linear (6,3) code with generator matrix 1 0 01 1 0 G- 0 1 0 0 1 1 a) Determine the parity check matrix H of the code. b) What is the minimum distance of the code? How many errors can this code correct and detect? c) Show the results in b) using decoding table d) Find the most likely codeword, given that the noisy received codeword is 010101. e) Now suppose 001101...
(c) Consider the (7, 4) Hamming Code defined by the generator polynomial g(x)-1 +x+x'. The code word 1000101 is sent over a noisy channel, producing the received word 0000101 that has a single error. Determine the syndrome polynomial s(x) for this received word. Find its corresponding message vector m and express m in polynomial m(x). 0
coding theory
1. If 100 031 Go 01 0 0 9 0001 27 01 0 054 is a generator matrix for a linear code over Fi encode the message stream m(2, 3,9, 6, 1,4,3, 8) (by breaking it into encodable pieces). 2. If T (1 0 1 2 3 4 be the transpose of a parity-check matrix for a perfect 1-error-correcting code over Fs, with implicit generator matrix 4410 0 0 3 4 0 1 0 0 1400 01
1....
1. Channel Coding We would like to add linear block code (3,6) using the generator matrix: 1 001 01 G-0 1 0 0 1 1 (a) (5 points) Determine the parity check matrix H (b) (20 points) What is the minimum distance of this code? How many error can this code correct? (c) (5 points) What is the code word for the data sequence 011000101111? (d) (20 points) If you receive the codeword 010001000010101010, what is the transmitted sequence?
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