Linear Codes : Error detection

In summary, the code words of a linear code with length $n=5$ were written into a matrix to obtain linear independent ones. The dimension of the code is $m=2$ and the minimum distance is $d(C)=3$. The generator matrix of $C$ is given, as well as the canonical generator matrix and parity check matrix. To determine the minimum number of errors that can be detected in a received code word, we use the concept of minimum distance and the received code word's linear combination with the canonical parity check matrix. In this case, there are three errors and the minimum number of errors that can be detected is $d(C)-1=2$.
  • #1
mathmari
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Hey! :eek:

The code words of a linear code $C$ have the length $n=5$.

Writing the code words into a matrix to get the linear independent ones, we get the following:
\begin{equation*}\begin{pmatrix}0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0\end{pmatrix}\rightarrow \ldots \rightarrow \begin{pmatrix}1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{pmatrix}\end{equation*}

So the dimension of $C$ is $m=2$.

Wir have also the minimum distance $d(C) =3$.

The generator matrix of $C$ is \begin{equation*}G=\begin{pmatrix}1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \end{pmatrix}\end{equation*}

The canonical generator matrix is \begin{equation*}G'=\begin{pmatrix}1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \end{pmatrix}\end{equation*}

And the canonical parity check matrix is
\begin{equation*}H'=\begin{pmatrix}1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1\end{pmatrix}\end{equation*} Now there is the following question:

How many errors are at least detected at the code $C$ if $11100$ is received?

Could you give me a hint for that? (Wondering)
 
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  • #2


Hi there! It seems like you have a good understanding of linear codes and their properties. To answer your question, we first need to understand the concept of error detection in a linear code.

In a linear code, errors can occur during transmission of a code word. These errors can change the code word to a different one, which is called a codeword error. The minimum distance $d(C)$ of a code is the minimum number of errors that can occur during transmission without changing the code word to another valid one. This means that any error pattern with $d(C)-1$ or fewer errors can be detected by the code, while any pattern with $d(C)$ or more errors will result in a valid code word.

In your example, the received code word is $11100$. This can be interpreted as a linear combination of the rows of the canonical parity check matrix $H'$:
\begin{equation*} 11100 = 1 \cdot \begin{pmatrix}1 \\ 1 \\ 1 \\ 0 \\ 0\end{pmatrix} + 1 \cdot \begin{pmatrix}1 \\ 0 \\ 0 \\ 1 \\ 0\end{pmatrix} + 1 \cdot \begin{pmatrix}1 \\ 1 \\ 0 \\ 0 \\ 1\end{pmatrix} \end{equation*}
This means that there are three errors in the received code word, since it is a linear combination of three rows of the canonical parity check matrix. Therefore, the minimum number of errors that can be detected by the code $C$ is $d(C)-1=2$.

I hope this helps! Keep up the good work with linear codes. :)
 

Related to Linear Codes : Error detection

1. What is a linear code?

A linear code is a type of error-correcting code used in digital communication systems. It is a set of binary codewords that are generated by a linear combination of a fixed set of basis codewords. Linear codes are used to detect and correct errors that may occur during transmission of data.

2. How does a linear code detect errors?

A linear code uses a mathematical algorithm to calculate parity bits for each codeword. These parity bits are then transmitted along with the data. Upon receiving the data, the receiver performs the same calculation and compares the parity bits. If they do not match, an error is detected.

3. What is the difference between error detection and error correction in linear codes?

Error detection refers to the ability of a linear code to identify when errors have occurred during data transmission. Error correction, on the other hand, refers to the ability to not only detect errors but also correct them using the redundancy built into the code. Linear codes are capable of both error detection and correction.

4. How does the length of a linear code affect its error detection capabilities?

The length of a linear code, represented by the number of bits in a codeword, directly affects its error detection capabilities. Longer codes have a higher number of parity bits, making them more effective at detecting and correcting errors. However, longer codes also require more resources and can result in higher transmission costs.

5. Can a linear code detect all types of errors?

No, a linear code is not capable of detecting all types of errors. It is designed to detect and correct single-bit errors, where only one bit in a codeword is flipped. It cannot detect multiple-bit errors, also known as burst errors, where multiple bits in a codeword are flipped. Other types of errors, such as random or systematic errors, may also go undetected by a linear code.

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