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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

Use maximum likelihood decoding to correct the following six-word coded message generated by a triple repetition code. Then decode the message.

101101101 110110101 110100101 101000111 110010011 011011011

To determine

To construct: The correct six-word coded message generated by a triple repetition code. Then, decode the message.

101101101110110101110100101101000111110010011011011011

Explanation

Given information:

101101101110110101110100101101000111110010011011011011

Formula used:

Maximum likelihood decoding: Multiple errors can be detected and corrected if each k bit word is mapped onto a 3k bit code word according to the following scheme:

x1x2x3...xkencodex1x2x3...xkx1x2x3...xkx1x2x3...xk

Explanation:

In the coded massage with k=3

If the 9 bit code word (for a 3 bit word) 101101101 is received, then an error is not detected.

By separating the code word into three equal parts

101101101

And comparing bit by bit, we note that all bits in each part are agreed.

Thus, error is not occur in 101101101.

If the 9 bit code word (for a 3 bit word) 110110101 is received, then an error is detected.

By separating the code word into three equal parts

110110101

And comparing bit by bit, we note that second and third bits in each part do not agree.

We correct the error by choosing the digit that occur most often.

For second bit, it is 1.

For the third bit, it is 0.

Thus, corrected code word is, 101101101.

If the 9 bit code word (for a 3 bit word) 110100101 is received, then an error is detected.

By separating the code word into three equal parts

110100101

And comparing bit by bit, we note that second and third bits in each part do not agree.

We correct the error by choosing the digit that occur most often.

In this case,

For second bit, it is 0.

For the third bit, it is 0

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