0

Now, we combine Eqs. (1.6) and (1.7) as shown in Eq. (1.8) to yield the desired matrix

In the computation shown above, symbol represents modulo-2 addition. As shown in Eq. (1.10), modulo-2 addition yields the bits

We now perform the computation of [

We notice that parity bits in code vectors

## Generator Matrix Example

**Construct a generator matrix and write all the code words of a (6, 3) LBC.**

**Solution:**Following the rules given above, we first construct

*G*as given below. From definition, we have. From definition, we have

[

*G*]_{k}_{ × n}= [*I*]_{k }_{× k}[*P*]_{ k }_{× n ‒ k }*(1.5)*_{ }
We now construct

*G*based on Eq. (1.5) and the rules given above and the result is shown in Eq. (1.6). In Eq. (1.6), we first construct the 3×3*I*matrix. As shown in Eq. (1.6), all the diagonal elements of*I*are**1**s; all of the rest are**0**s.
After constructing

*I*, we proceed to form the*P*(parity) matrix as a*k*× (*n*‒*k*) = 3 × 3 matrix. In this matrix, no column must have a single-**1**entry and no column must have all-**0**entries. Obeying this rule, we have as the first-column elements of*P*the set of numbers**110**. For the second column, we have chosen**101**, and for the third column, we have**111**. Equation (1.6) gives [*P*].Now, we combine Eqs. (1.6) and (1.7) as shown in Eq. (1.8) to yield the desired matrix

*G*.

With

*k*= 3, the data vectors are the binary numbers from**000**to**111**. Of these, it is quite common to avoid the combination of**000**. The rest of the data vectors will give finite code words by the multiplication of*D*and*G*as per the rules of Eq. (1.1). Designating data vectors*D*_{1}=**001**,*D*_{2}=**010**, and so on, we now compute code vectors*C*_{1},*C*_{2}, and*C*_{7}. Let us first determine the code word corresponding to the data vector*D*_{1}= 001. For this, we write Eq. (1.9) as given below:
[

We perform matrix multiplications as a *C*_{1}] = [*D*_{1}][*G*]*row*×

*column*operation and then add the resulting bits using

*modulo*-2 addition. It may be remembered that since the fist 3 elements in the

*G*matrix are the identity elements, and hence the multiplication of [

*D*] × [

*I*] = [

*D*] itself. So in all these cases, we only need to perform the [

*D*] × [

*P*] operation. Thus, we get

In the computation shown above, symbol represents modulo-2 addition. As shown in Eq. (1.10), modulo-2 addition yields the bits

**0**,

**1**, and

**1**, which are the parity bits. The resulting code word

*C*

_{1}is shown in the combined form using bold fonts as

**001011**. In a similar way, we find code vector

*C*

_{2}:

We now perform the computation of [

*C*

_{7}] = [

*D*

_{7}][

*G*]. Thus

*C*

_{1},

*C*

_{2}and

*C*

_{7}are, respectively,

**011**,

**101**, and

**001**. We also notice that parity word 0

**11**is always associated with data word

**001**. Similarly, parity word

**101**is always associated with

**010**, and

**001**is always associated with

**111**. We thus find that to each of the data word, a unique set of parity bits are added and, as stated earlier, it is this uniqueness that helps in the recovery of original data from noise-corrupted data. It may be noted that code vectors

*C*

_{3}to

*C*

_{7}can be similarly determined by using the respective data vectors and the

*G*matrix. The reader is advised to determine these code vectors.

### This post was written by: Sreejith Hrishikeshan

Sreejith Hrishikesan Nair is a M-Tech graduate in Communication Systems. He completed B-tech Degree in Electronics and Communication.He is a person who wants to implement new ideas in the field of Technology.

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