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,

  $\left[16125\right],\left[1195\right],\left[0195\right],\left[1440\right],\left[131514\right],\left[5250\right]$

Explanation of Solution

Given information:

Given massage

PLEASE SEND MONEY

Given matrix

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

Calculation:

First create the $1\times 3$ encoding matrix.

We split the massage into groups two:

“P L E” “A S E” “_ S E” “N D_” “M O N” “E Y_”

16 12 5 1 19 5 0 19 5 14 4 0 13 15 14 5 25 0

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

  $\left[16125\right],\left[1195\right],\left[0195\right],\left[1440\right],\left[131514\right],\left[5250\right]$

(b)

To determine

To encode: the massage using the encoding matrix.

Answer to Problem 52E

The following cryptogram,

-43 6 9 -38 -45 -13 -42 -47 -14 44 16 10 49 9 12 -55 -65 -20.

Explanation of Solution

Calculation:

Now to get the coded row matrices,

Uncoded matrix Encoding matrix coded matrix

  $\begin{array}{c}\left[16125\right]\left[\begin{array}{ccc}4& 2& 1\\ -3& -3& -1\\ 3& 2& 1\end{array}\right]\left[4369\right]\\ \left[1195\right]\left[\begin{array}{ccc}4& 2& 1\\ -3& -3& -1\\ 3& 2& 1\end{array}\right]\left[-38-4513\right]\\ \left[0195\right]\left[\begin{array}{ccc}4& 2& 1\\ -3& -3& -1\\ 3& 2& 1\end{array}\right]\left[-42-47-14\right]\\ \left[1440\right]\left[\begin{array}{ccc}4& 2& 1\\ -3& -3& -1\\ 3& 2& 1\end{array}\right]\left[441610\right]\\ \left[131514\right]\left[\begin{array}{ccc}4& 2& 1\\ -3& -3& -1\\ 3& 2& 1\end{array}\right]\left[49912\right]\\ \left[5250\right]\left[\begin{array}{ccc}4& 2& 1\\ -3& -3& -1\\ 3& 2& 1\end{array}\right]\left[-55-65-20\right]\end{array}$

So the sequence of coded row matrices

  $\left[-682135\right],\left[-661439\right],\left[4369\right],\left[-38-45-13\right],\left[-42-47-14\right],\left[441616\right]\left[49912\right]\left[-55-65-20\right]$

Finally removing the matrix notation produces the following cryptogram,

43 6 9 -38 -45 -13 -42 -47 -14 44 16 10 49 9 12 -55 -65 -20.