# Binary encoding function problem

• sara15
In summary, the binary encoding function is a linear transformation using a 3x7 transformation matrix. The parity check matrix can be obtained by using the bottom four rows of the transformation matrix for error detection. The minimum distance of the code refers to the shortest distance between any two possible output vectors. To decode a given vector, you would need to determine the closest output vector and use its corresponding values for a1, a2, and a3.
sara15
consider the binary encoding function that sends (a1,a2,a3) into (a1,a2,a3,a1+a2, a2+a3, a1+a3,a1+a2+a3). First , how can I give a generator matrix of this code
. then how can I provide the parity check matrix , giving n,k and the minimum distance of the code. Finall, how to decode m1=1100010 and m2= 0111010

sara15 said:
consider the binary encoding function that sends (a1,a2,a3) into (a1,a2,a3,a1+a2, a2+a3, a1+a3,a1+a2+a3). First , how can I give a generator matrix of this code
. then how can I provide the parity check matrix , giving n,k and the minimum distance of the code. Finall, how to decode m1=1100010 and m2= 0111010

Hey there sara and welcome to the forums.

Since the transformation you are looking for is a linear transformation, we know that we can use linear algebra (ie a matrix).

So we have a three dimensional input and we want to generate a seven dimensional output.

So let's assign our vector as a 1x3 (row vector) and our output as a 1x7 (column vector).

So we have to compose a transformation that goes from
1x3 x (axb) = 1x7

This means that our transformation matrix has dimensions 3x7.

Using this information and the properties of matrix multiplication can you now populate your 3x7 matrix with values?

Thank you very much

ok I could get the generator matrix , but I still do not know how to find the parity check matrix and the minimum distance and then decode m1= 1100010 and m2 = 0111010
Many thanks
Sara

What kind of codes are you dealing with?

The older parity codes only used one bit for the block, but it seems here you are using four bits for the error correction. So two questions?

1) What is the error correction code mechanism you are using?
2) Is the distance you are referring to called the "Hamming distance"?

$$\left[ \begin{array} {c c c} 1&0&0\\0&1&0\\0&0&1\\ 1&1&0\\0&1&1\\1&0&1\\1&1&1\end{array} \right]$$
The bottom four rows are used for error detection.

Since you're dealing with binary values, there are eight possible values for an input (a1, a2, a3), so there are eight possible output vectors.

I would guess that the minimum distance you mentioned is the shortest distance between any two of the eight possible output vectors. I'm not sure that this is what you're supposed to do, as it would entail finding the distance between all pairs of the eight vectors, which would be pretty tedious.

On the other hand, maybe what you're supposed to do for the two vectors you are supposed to decode, is to find which of the eight output vectors is the closest. So for example, if the vector you're supposed to decode is (1,1,0,0,0,1,0), and you determine that the closest vector is (1,1,0,0,1,1,0), then the latter would be the corrected vector, which you could decode as a1 = 1, a2 = 1, and a3 = 0.

## 1. What is a binary encoding function?

A binary encoding function is a mathematical operation that converts data into binary code, which is a string of 0s and 1s. It is commonly used in computer science to represent text, numbers, and other data in a format that can be understood and processed by computers.

## 2. How does a binary encoding function work?

A binary encoding function takes in a set of data and assigns a unique binary code to each piece of information. This is typically done by converting the data into its binary representation, and then combining the individual binary digits into a single string. The resulting string is the binary code for that particular piece of data.

## 3. What is the purpose of using a binary encoding function?

The main purpose of using a binary encoding function is to efficiently store and transmit data in a form that can be easily processed by computers. Binary code takes up less space than other data formats, making it ideal for storing and transferring large amounts of data. It also allows for quick and accurate data retrieval and manipulation.

## 4. Are there different types of binary encoding functions?

Yes, there are several different types of binary encoding functions, each with its own specific purpose and method. Some examples include ASCII, Unicode, and Base64 encoding. Each type is used for different types of data and has its own set of rules and algorithms for converting data into binary code.

## 5. What are some common challenges with binary encoding functions?

One common challenge with binary encoding functions is ensuring that the data is accurately converted and decoded. This can be especially tricky when dealing with large or complex data sets. Additionally, different types of binary encoding functions may have different limitations and compatibility issues, which can make it difficult to transfer and interpret data between different systems.

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