Question

Show that the linear code C with parity-check matrix $\left(\begin{array}{cc}1& 0\\ 0& 1\\ 1& 0\\ 0& 1\\ 1& 0\\ 0& 1\end{array}\right)$ cannot correct every single error.

Short answer

Every single error cannot be corrected by a linear code with parity-check matrix .

Step-by-step solution

Conceptual Introduction

Coding theory is the study of a code's characteristics and how well-suited it is to various applications. Data compression, cryptography, error detection, and correction, as well as data transport and storage, all depend on codes.

Introduction

Assume C is a linear code with H as the parity-check matrix. Parity-check matrix decoding corrects all single errors if every row of H is nonzero and no two are the same. Consider a linear code C with following parity-check matrix:

$\mathrm{H}=\left(\begin{array}{cc}1& 0\\ 0& 1\\ 1& 0\\ 0& 1\\ 1& 0\\ 0& 1\end{array}\right)$

The purpose is to show every single error cannot be corrected by a linear code C with parity-check matrix H.

Analyzing and proving the required statement

Now, given parity-check matrix is;

$\mathrm{H}=\left(\begin{array}{cc}1& 0\\ 0& 1\\ 1& 0\\ 0& 1\\ 1& 0\\ 0& 1\end{array}\right)$

Clearly, all the rows have all-nonzero entry, but the entries in 1^st, 3^rd and 5^th row are same.

Also, the entries in 2^nd, 4^th and 6^th row are same.

Thus, according to the theorem stated above, every single error cannot be corrected by a linear code C with parity-check matrix H.