Coding Theory Mock 1

Question 1

Let $C$ be binary linear code with the following generator matrix…

$$G=\left[\begin{array}{ccccc}1& 1& 1& 1& 0\\ 1& 0& 1& 0& 1\end{array}\right]$$

Question 1a

List elements of a code based on a generator matrix and determine its parameters.

$[5,2,3]$ - to-do: neaten the method I wrote in my notebook, after checking the answer.

Question 1b

Find a standard matrix and a parity-check matrix linearly equivalent to a code.

$$\left[\begin{array}{ccccc}1& 0& 1& 0& 0\\ 0& 1& 0& 1& 0\\ 1& 1& 0& 0& 1\end{array}\right]$$

to-do: neaten the method I wrote in my notebook, after checking the answer.

Question 1c

Calculate the probability that errors over transmissions are not detected, as a polynomial.

$$P(\text{undetected error})=\sum _{i=1}^n{A}_i{p}^i(1-p{)}^{n-i}$$ $$p^5-3p^4+2p^3$$