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Question: 5]b) Using the encoding matrix $ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ encode and decode the message I*LOVE*MUMBAI.
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Subject:- Applied Mathematics

Marks:- 3

Mumbai Unversity>FE>Sem1>Applied Maths1

m1(81) • 11 views
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modified 4 days ago  • written 4 days ago by gravatar for Mayank Aggarwal Mayank Aggarwal0
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A B C D E F G H I J K L M N O P Q R S T U V W X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

* Y. Z

27.25. 26

I * LOVEMUMBAI

9 27 12 15 22 5 27 13 21 13 2 1 9 27

Encoding the message includes the following process.

$ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $$ \begin{bmatrix} 9 & 12 & 22 & 27 & 12 & 2 & 9 \ 27 & 15 & 5 & 13 & 13 & 1 & 27 \end{bmatrix} $ = $ \begin{bmatrix} 9+27 & 12+15 & 22+5 & 27+13 & 12+13 & 2+1 & 9+27 \ 27 & 15 & 5 & 13 & 13 & 1 & 27 \end{bmatrix} $ = $ \begin{bmatrix} 36 & 27 & 27 & 40 & 25 & 3 & 36 \ 27 & 15 & 5 & 13 & 13 & 1 & 27 \end{bmatrix} $ Hence the encoded message we get as , 36, 27, 27, 15, 27, 5, 40, 13, 25, 13, 3, 1, 36, 27. Now the process of decoding is as follows. Inverse of $ \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} $ is $ \begin{bmatrix} 1 & -1 \ 0 & 1 \end{bmatrix} $ $ \begin{bmatrix} 1 & -1 \ 0 & 1 \end{bmatrix} $ $ \begin{bmatrix} 36 & 27 & 27 & 40 & 25 & 3 & 36 \\ 27 & 15 & 5 & 13 & 13 & 1 & 27 \end{bmatrix} $

$ \begin{bmatrix} 9 & 12 & 22 & 27 & 12 & 2 & 9 \\ 27 & 15 & 5 & 13 & 13 & 1 & 27 \end{bmatrix} $

Hence the original message is obtained again after encoding and decoding.

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written 4 days ago by gravatar for Mayank Aggarwal Mayank Aggarwal0
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