Question

Construct a (5, 2) code that corrects single errors.

Short answer

The $\left(5,2\right)$ codes that correct single errors are:$00000, 01010, 10111, 11101$

Step-by-step solution

Theorem 16.6 and Corollary 16.4

Theorem 16.6

If G is a $k\times n$ standard generator matrix, then $\left\{uG|u\in B\left(k\right)\right\}$ is a systematic $\left(n,k\right)$ code

Corollary 16.4

A linear code detects 2t errors and corrects terrors if and only if the Hamming weight of every nonzero codeword is at least $2t+1$.

Constructing a (5, 2) code that corrects single errors

According to Theorem 16.6 to construct a $\left(5,2\right)$code, the dimension of the generator matrix should be $2\times 5$

Let the generator matrix,

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

The message words of the code $\left(5,2\right)$are:

$00, 01, 10, 11$

Therefore, to get the code words we must multiply them with the generator

$$\begin{gathered} \left(00\right)\left[\begin{array}{ccccc}1& 0& 1& 1& 1\\ 0& 1& 0& 1& 0\end{array}\right]=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\end{array}\right] \\ \left(01\right)\left[\begin{array}{ccccc}1& 0& 1& 1& 1\\ 0& 1& 0& 1& 0\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& 1& 0\end{array}\right] \\ \left(10\right)\left[\begin{array}{ccccc}1& 0& 1& 1& 1\\ 0& 1& 0& 1& 0\end{array}\right]=\left[\begin{array}{ccccc}1& 0& 1& 1& 1\end{array}\right] \\ \left(11\right)\left[\begin{array}{ccccc}1& 0& 1& 1& 1\\ 0& 1& 0& 1& 0\end{array}\right]=\left[\begin{array}{ccccc}1& 1& 1& 0& 1\end{array}\right] \end{gathered}$$

Therefore, the code words for the given matrix generator are:

$00000, 01010, 10111, 11101$

Now we see the humming weight of nonzero codewords is at least $3=2\times 1+1$

Therefore, from corollary 16.4, with $t=1$,code generated by the matrix G corrects single error.

Hence $\left(5,2\right)$ code that corrects single errors are: $00000, 01010, 10111, 11101$