$2.(24$ pts$)$ Single parity check code and the related product code

A $($k$+1,$k$)$ or $($n$,$n$-1)$ single parity check code is created by simply adding a single bit at the end that is the exclusive$-$or of all the k message bits. We can also consider the $(($k$+1)$x$($k$+1),$ k$^2)$ product code for which each of the k $+1$ rows and k $+1$ columns is a single parity check code, which looks like this: