he Modified Armstrong$-$Kirikeza Method.

On $12$th September, Messrs. Armstrong and Kirikeza$2$ came up with

the following method for detecting and correcting single$-$digit errors:

Given a codeword, we will create a message by following the following

steps:

Step $1$: Write the codeword twice.

Step $2$: Append a $0$ to the second codeword if the number of $1$s in the

codeword is odd. Otherwise, append a $1.$

Therefore, if the codeword was $0110,$ the message would be $011001101.$

Here is a modification to their method:

Step $1$m: Write the codeword three times.

Step $2$m: Append a $0$ to the second and third codewords if the number

of $1$s in the codeword is odd. Otherwise, append a $1.$

So if the codeword was $1011,$ the message is $10111011010110,$ or $1011$

$1011010110($with spaces added for readability$).$

$2$Feel free to blame them for this question.

$5$

$($a$)$ Prove that this method detects and corrects any two errors $($as$-$

suming the codeword is length $4).$

$($b$)$ Explain why the efficiency is about $28\mathrm{.}6\%.($Again$,$ assuming the

codeword is length $4.)$

$($c$)$ What happens to the efficiency of this method as the length of

the codeword goes to infinity?