Single-Parity-Check Codes What are single parity-check(SPC) codes?
I know about Repetition codes and generator matrix for them but have not been able to find much information about the SPC on the internet. Can anyone suggest ways to find its generator matrix?
Any suggested reading will be helpful. Thanks in advance!
 A: The SPC is a linear block code with a single parity check digit(added in the end to make the codeword).
Let the information bit be given as $$b_1,b_2,b_3 ...,b_k$$ then the check bit will be $$b_{k+1} = b_1 + b_2 + b_3 + ... + b_k $$
and the codeword will be $$b_1,b_2,b_3 ...,b_k,b_{k+1}$$
At the receiver end detector will be able to detect the error if and only if even(odd) parity is used and there are even(odd) numbers of 1's in the received signal. In all other cases, the error will go undetected.
They have a generator matrix of $N \times N-1$ dimension. I will try to show it using an example with a code of length $N = 4$.
In our case the generator matrix will look like $$
   G=
  \left[ {\begin{array}{cccc}
   1 & 0 & 0 & 1\\
   0 & 1 & 0 & 1 \\
   0 & 0 & 1 & 1\\
  \end{array} } \right]
$$
And thus the Parity matrix is
$$ H=
  \left[ {\begin{array}{cccc}
   1 & 1 & 1 & 1\\
  \end{array} } \right]$$
It can be seen that in general, the parity matrix will always be a Row matrix of all 1 elements and thus $d_{min}$ of the SPC is always 2.
