In this question, we are going to construct a (7, 3) linear block code by extending a (6, 3) linear block code. The generator matrix of the (6, 3) code is 1001 1 0 G= 0 1 0 1 0 1 0 0 1 0 1 1 The (7, 3) code is extended from the (6, 3) code by appending one extra bit to into the codeword such that the Hamming weight of the resulting codeword is even. For example, for an input information se- quence of [111], the (6, 3) code produces an codeword of [111000] which is then extended by appending an extra bit "1" at the end to form the final codeword of [1110001]. The reason for appending a bit "1" (instead of bit "0") is to make the Hamming weight of the resulting codeword in this example be an even number. Another way to describe the (7, 3) code would be that it consists of a (6, 3) code with generator matrix G followed by another single parity check code with even parity.
In this question, we are going to construct a (7, 3) linear block code by extending a (6, 3) linear block code. The generator matrix of the (6, 3) code is 1001 1 0 G= 0 1 0 1 0 1 0 0 1 0 1 1 The (7, 3) code is extended from the (6, 3) code by appending one extra bit to into the codeword such that the Hamming weight of the resulting codeword is even. For example, for an input information se- quence of [111], the (6, 3) code produces an codeword of [111000] which is then extended by appending an extra bit "1" at the end to form the final codeword of [1110001]. The reason for appending a bit "1" (instead of bit "0") is to make the Hamming weight of the resulting codeword in this example be an even number. Another way to describe the (7, 3) code would be that it consists of a (6, 3) code with generator matrix G followed by another single parity check code with even parity.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![In this question, we are going to construct a (7,3) linear block code by extending a (6, 3) linear block
code. The generator matrix of the (6, 3) code is
G =
1
0 0 1 1 0
0 1 0 1 0 1
00101 1
The (7, 3) code is extended from the (6, 3) code by appending one extra bit to into the codeword such
that the Hamming weight of the resulting codeword is even. For example, for an input information se-
quence of [111], the (6, 3) code produces an codeword of [111000] which is then extended by appending
an extra bit "1" at the end to form the final codeword of [1110001]. The reason for appending a bit "1"
(instead of bit "0") is to make the Hamming weight of the resulting codeword in this example be an
even number.
Another way to describe the (7, 3) code would be that it consists of a (6, 3) code with generator matrix
G followed by another single parity check code with even parity.
a) What is the code rate of this (7,3) code?
b) What is the generator matrix of this (7, 3) code?
c) What is the minimum Hamming distance of this (7,3) code?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bc6d303-52a6-496d-a490-2e6b9ca359c2%2F1d337250-93ff-4fe9-9b9c-c7eff7a48788%2Fgq5zdp_processed.png&w=3840&q=75)
Transcribed Image Text:In this question, we are going to construct a (7,3) linear block code by extending a (6, 3) linear block
code. The generator matrix of the (6, 3) code is
G =
1
0 0 1 1 0
0 1 0 1 0 1
00101 1
The (7, 3) code is extended from the (6, 3) code by appending one extra bit to into the codeword such
that the Hamming weight of the resulting codeword is even. For example, for an input information se-
quence of [111], the (6, 3) code produces an codeword of [111000] which is then extended by appending
an extra bit "1" at the end to form the final codeword of [1110001]. The reason for appending a bit "1"
(instead of bit "0") is to make the Hamming weight of the resulting codeword in this example be an
even number.
Another way to describe the (7, 3) code would be that it consists of a (6, 3) code with generator matrix
G followed by another single parity check code with even parity.
a) What is the code rate of this (7,3) code?
b) What is the generator matrix of this (7, 3) code?
c) What is the minimum Hamming distance of this (7,3) code?
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