4. The generator matrix for a (7,3) code is: 0 1 1 G=1 0 1 1 0 1 10 10 0 1 1 0 0 1 a. A vector c= (x1, x2, ..., x7) is a valid codeword if and only if it satisfies the equation He" = 0 for some matrix H. List all codewords of this code. b. What is the rate of the code? с. What is the weight of the code?
4. The generator matrix for a (7,3) code is: 0 1 1 G=1 0 1 1 0 1 10 10 0 1 1 0 0 1 a. A vector c= (x1, x2, ..., x7) is a valid codeword if and only if it satisfies the equation He" = 0 for some matrix H. List all codewords of this code. b. What is the rate of the code? с. What is the weight of the code?
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![4. The generator matrix for a (7,3) code is:
o 1 1 1 1 0 O
G=1 0 1 1 0 1 0
1 10 10 0
a. A vector e = (x1, x2, ..., x7) is a valid codeword if and only if it satisfies the equation Hc" = 0
1
for some matrix H. List all codewords of this code.
b. What is the rate of the code?
c. What is the weight of the code?
d. What is the Hamming distance of the code? What is the maximum number of errors that can
be detected/corrected? You may use the syndrome table below to help you.
If you receive the codeword r = [0 0 1 1 0 1 1] then what is the syndrome for this codeword?
е.
If the syndrome decoding table is below, find the transmitted message signal.
Syndrome
0000
1000
0100
Error
0000000
1000000
0100000
0010000
0010
0001000
0001
0000100
0111
0000010
1011
0000001
1101
1100000
1100
1010000
1010
0110000
0110
1001000
1001
0101000
0101
0011000
0011
1111
1110
1000100
1110000
If the syndrome of the received codeword is 1000 then only the 1st bit is in error, if the
syndrome is 0100 then the 2nd bit is in error, etc.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0621effe-d84c-4759-a682-0d93b5d429e4%2Fb6e53097-8d0c-498d-8072-389dfb8a2147%2F0pcucd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. The generator matrix for a (7,3) code is:
o 1 1 1 1 0 O
G=1 0 1 1 0 1 0
1 10 10 0
a. A vector e = (x1, x2, ..., x7) is a valid codeword if and only if it satisfies the equation Hc" = 0
1
for some matrix H. List all codewords of this code.
b. What is the rate of the code?
c. What is the weight of the code?
d. What is the Hamming distance of the code? What is the maximum number of errors that can
be detected/corrected? You may use the syndrome table below to help you.
If you receive the codeword r = [0 0 1 1 0 1 1] then what is the syndrome for this codeword?
е.
If the syndrome decoding table is below, find the transmitted message signal.
Syndrome
0000
1000
0100
Error
0000000
1000000
0100000
0010000
0010
0001000
0001
0000100
0111
0000010
1011
0000001
1101
1100000
1100
1010000
1010
0110000
0110
1001000
1001
0101000
0101
0011000
0011
1111
1110
1000100
1110000
If the syndrome of the received codeword is 1000 then only the 1st bit is in error, if the
syndrome is 0100 then the 2nd bit is in error, etc.
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