**Error Detection in Computer Networks-**

Error detection is a technique that is used to check if any error occurred in the data during the transmission. |

Some popular error detection methods are-

- Single Parity Check
- Cyclic Redundancy Check (CRC)
- Checksum

In this article, we will discuss about Cyclic Redundancy Check (CRC).

**Cyclic Redundancy Check-**

- Cyclic Redundancy Check (CRC) is an error detection method.
- It is based on binary division.

**CRC Generator-**

- CRC generator is an algebraic polynomial represented as a bit pattern.
- Bit pattern is obtained from the CRC generator using the following rule-

The power of each term gives the position of the bit and the coefficient gives the value of the bit. |

**Example-**

Consider the CRC generator is x^{7} + x^{6} + x^{4} + x^{3} + x + 1.

The corresponding binary pattern is obtained as-

Thus, for the given CRC generator, the corresponding binary pattern is 11011011.

**Properties Of CRC Generator-**

The algebraic polynomial chosen as a CRC generator should have at least the following properties-

**Rule-01:**

- It should not be divisible by x.
- This condition guarantees that all the burst errors of length equal to the length of polynomial are detected.

**Rule-02:**

- It should be divisible by x+1.
- This condition guarantees that all the burst errors affecting an odd number of bits are detected.

**Important Notes-**

If the CRC generator is chosen according to the above rules, then-

- CRC can detect all single-bit errors
- CRC can detect all double-bit errors provided the divisor contains at least three logic 1’s.
- CRC can detect any odd number of errors provided the divisor is a factor of x+1.
- CRC can detect all burst error of length less than the degree of the polynomial.
- CRC can detect most of the larger burst errors with a high probability.

**Steps Involved-**

Error detection using CRC technique involves the following steps-

**Step-01: Calculation Of CRC At Sender Side-**

At sender side,

- A string of n 0’s is appended to the data unit to be transmitted.
- Here, n is one less than the number of bits in CRC generator.
- Binary division is performed of the resultant string with the CRC generator.
- After division, the remainder so obtained is called as
**CRC**. - It may be noted that CRC also consists of n bits.

**Step-02: Appending CRC To Data Unit-**

At sender side,

- The CRC is obtained after the binary division.
- The string of n 0’s appended to the data unit earlier is replaced by the CRC remainder.

**Step-03: Transmission To Receiver-**

- The newly formed code word (Original data + CRC) is transmitted to the receiver.

**Step-04: Checking at Receiver Side-**

At receiver side,

- The transmitted code word is received.
- The received code word is divided with the same CRC generator.
- On division, the remainder so obtained is checked.

The following two cases are possible-

**Case-01: Remainder = 0**

If the remainder is zero,

- Receiver assumes that no error occurred in the data during the transmission.
- Receiver accepts the data.

**Case-02: Remainder ≠ 0**

If the remainder is non-zero,

- Receiver assumes that some error occurred in the data during the transmission.
- Receiver rejects the data and asks the sender for retransmission.

**Also Read-****Parity Check**

**PRACTICE PROBLEMS BASED ON CYCLIC REDUNDANCY CHECK (CRC)-**

**Problem-01:**

A bit stream 1101011011 is transmitted using the standard CRC method. The generator polynomial is x^{4}+x+1. What is the actual bit string transmitted?

**Solution-**

- The generator polynomial G(x) = x
^{4}+ x + 1 is encoded as 10011. - Clearly, the generator polynomial consists of 5 bits.
- So, a string of 4 zeroes is appended to the bit stream to be transmitted.
- The resulting bit stream is 1101011011
**0000**.

Now, the binary division is performed as-

From here, CRC = 1110.

Now,

- The code word to be transmitted is obtained by replacing the last 4 zeroes of 1101011011
**0000**with the CRC. - Thus, the code word transmitted to the receiver = 1101011011
**1110**.

**Problem-02:**

A bit stream 10011101 is transmitted using the standard CRC method. The generator polynomial is x^{3}+1.

- What is the actual bit string transmitted?
- Suppose the third bit from the left is inverted during transmission. How will receiver detect this error?

**Solution-**

**Part-01:**

- The generator polynomial G(x) = x
^{3}+ 1 is encoded as 1001. - Clearly, the generator polynomial consists of 4 bits.
- So, a string of 3 zeroes is appended to the bit stream to be transmitted.
- The resulting bit stream is 10011101
**000**.

Now, the binary division is performed as-

From here, CRC = 100.

Now,

- The code word to be transmitted is obtained by replacing the last 3 zeroes of 10011101
**000**with the CRC. - Thus, the code word transmitted to the receiver = 10011101
**100**.

**Part-02:**

According to the question,

- Third bit from the left gets inverted during transmission.
- So, the bit stream received by the receiver = 10111101100.

Now,

- Receiver receives the bit stream = 10111101100.
- Receiver performs the binary division with the same generator polynomial as-

From here,

- The remainder obtained on division is a non-zero value.
- This indicates to the receiver that an error occurred in the data during the transmission.
- Therefore, receiver rejects the data and asks the sender for retransmission.

To watch video solution, click **here**.

To gain better understanding about Cyclic Redundancy Check,

**Next Article-****Checksum**

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