Chapter 8.5, Problem 51E

(a)

To determine

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

Answer to Problem 51E

Uncoded matrices for the given massage are,

  $\left[8116\right],\left[16250\right],\left[2918\right],\left[2084\right],\left[1250\right]$

Explanation of Solution

Given information:

Given massage

HAPPY BIRTHDAY

Given matrix

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

Calculation:

We have to first the $1\times 3$ encoding matrix.

We split the massage into groups two:

“H A P” “P Y _” “B I R” “T H D” “A Y_”

8 1 16 16 25 0 2 9 18 20 8 4 1 25 0

Thus the set of uncoded matrices for the given massage are,

  $\left[8116\right],\left[16250\right],\left[2918\right],\left[2084\right],\left[1250\right]$

(b)

To determine

To encode: the massage using the encoding matrix.

Answer to Problem 51E

-5 -41 -87 91 207 257 11 -5 41 40 80 84 76 117 227

Explanation of Solution

Calculation:

Now to get the coded row matrices,

Uncoded matrix Encoding matrix coded matrix

  $\begin{array}{c}\left[8116\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\left[-5-41-87\right]\\ \left[16250\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\left[91207257\right]\\ \left[2918\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\left[11-5-41\right]\\ \left[2084\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\left[408084\right]\\ \left[1250\right]\left[\begin{array}{ccc}1& 2& 2\\ 3& 7& 9\\ -1& -4& -7\end{array}\right]\left[76117227\right]\end{array}$

So the sequence of coded row matrices

  $\left[-5-41-87\right],\left[91207257\right],\left[11-5-41\right],\left[408084\right],\left[768084\right],\left[76117227\right]$

Finally removing the matrix notation produces the following cryptogram,

-5 -41 -87 91 207 257 11 -5 41 40 80 84 76 117 227