Finding the overhead and distance of an unknown code based on message making algorithm
The question is still pretty unclear, but I will provide a few hints and hope that they point you in the right direction.

The only code I'm aware of that can detect only one error and do nothing else is the repetition code \$R_2\$. This code takes a data bit \$b\$ and produces the codeword \$bb\$.

(Note that adding a parity bit can do more than detecting one error; it can detect any odd number of errors).

Now, by producing a code word that is a concatenation of \$A\$ with itself, all you're doing is using the same \$R_2\$ code again.

In summary, the mistery code is \$R_2\$, and the second code (which duplicates the word \$A\$) is just \$R_2\$ again.

The concatenation of two \$R_2\$ codes is equivalent to an \$R_4\$, where each bit is repeated four times; in other words, if the input is \$b\$, the output is \$bbbb\$. This code has distance \$4\$, can correct one error, and it can detect up to three errors.

Its code rate is \$1/4\$; it transmits four code bits per every data bit. The overhead then is \$3/4\$, or \$75\%\$.