Chapter 8.5, Problem 52E

(a)

To determine

To write: the uncoded $1\times 3$ row matrices for the massage.

Answer to Problem 52E

Uncoded matrices for the given massage are,

  $\left[15165\right],\left[18120\right],\left[9514\right],\left[01522\right],\left[51822\right],\left[014\right]$

Explanation of Solution

Given information:

Given massage

OPERATION OVERLOAD

Given matrix

  $A=\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]$

Calculation:

The message to be encoded is:

OPERATION OVERLOAD

Partition the message including blank spaces into groups of three.

Thus, the uncoded row matrices will be,

  $\left[15165\right],\left[18120\right],\left[9514\right],\left[01522\right],\left[51822\right],\left[014\right]$

(b)

To determine

To encode: the massage using the encoding matrix.

Answer to Problem 52E

58 122 149 1 -37 -95 10 -3 -45 23 17 -19 47 70 78 -1 -9 -19

Explanation of Solution

Calculation:

The coded row matrices are obtained by multiplying each of the uncoded row matrices by the matrix $A$ , as follows:

Uncoded Matrix encoding matrix coded matrix

  $\begin{array}{c}\left[15165\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[58122149\right]\\ \left[18120\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[1-37-95\right]\\ \left[9514\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[10-3-45\right]\\ \left[0522\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[2317-19\right]\\ \left[51812\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[477078\right]\\ \left[014\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]=\left[-1-9-19\right]\end{array}$

So the sequence of coded row matrices

  $\left[58122149\right],\left[1-37-95\right],\left[10-3-45\right],\left[2317-19\right],\left[4717-19\right],\left[477078\right]\left[-1-9-19\right]$

Finally removing the matrix notation produces the following cryptogram,

58 122 149 1 -37 -95 10 -3 -45 23 17 -19 47 70 78 -1 -9 -19