Question

Is there a $\left(6,2\right)$ code in which every nonzero codeword has Hamming weight at least 4?

Short answer

Yes, there is$\left(6,2\right)$ code in which every nonzero codeword has a Humming weight at least 4.

Step-by-step solution

Theorem 16.6 and definition of humming weight

• Theorem 16.6

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

• Humming weight

The Hamming weight of an element uof$\mathbf{B}\left(n\right)$. Is the number of nonzero coordinates in u; it is denoteddata-custom-editor="chemistry" $\mathbf{Wt}\left(u\right)$.

Constructing a (6,2) code

Since we have to construct $\left(6,2\right)$code, therefore according to the Theorem 16.6 dimension of generator matrix should be $2\times 6$

Let the generator matrix is

$G=\left(\begin{array}{cccccc}1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 0\end{array}\right)$

The message words of the code $\left(6,2\right)$are:

$00,01,10,11$

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

localid="1660539625994" $$\begin{gathered} \left(\begin{array}{cc}0& 0\end{array}\right)\left(\begin{array}{cccccc}1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 0\end{array}\right)=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right) \\ \left(\begin{array}{cc}0& 1\end{array}\right)\left(\begin{array}{cccccc}1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 0\end{array}\right)=\left(\begin{array}{cccccc}0& 1& 0& 1& 1& 0\end{array}\right) \\ \left(\begin{array}{cc}1& 0\end{array}\right)\left(\begin{array}{cccccc}1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 0\end{array}\right)=\left(\begin{array}{cccccc}1& 0& 1& 1& 1& 1\end{array}\right) \\ \left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{cccccc}1& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 1& 0\end{array}\right)=\left(\begin{array}{cccccc}1& 1& 1& 0& 0& 1\end{array}\right) \end{gathered}$$

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

localid="1660539641978" $000000,010110,101111,111001$

From above we can see that every nonzero code word has a humming weight of at least 4.

Hence, we can conclude, that there is a $\left(6,2\right)$code in which every nonzero codeword has a Humming weight of at least 4.