CRC and Burst Errors

CRC (Cyclic Redundancy Check) is an error detection technique used in digital networks and storage systems. It involves the use of a generator polynomial to compute a checksum for a given data set. This checksum, also known as the CRC code, is appended to the data and transmitted along with it. At the receiving end, the CRC code is recalculated and compared with the received CRC code to check for errors.

A burst error refers to a sequence of consecutive bits that are corrupted during transmission. The length of a burst error is defined as the number of consecutive bits that are affected by the error. The goal of CRC is to detect these burst errors and ensure the integrity of the transmitted data.

Generator Polynomial and Degree

The generator polynomial, denoted as G(x), is a key component of the CRC algorithm. It is a binary polynomial of degree m, where m is the length of the checksum (CRC code). The degree of a polynomial is the highest power of x in the polynomial.

Detecting Bursts of up to m Errors

To detect a burst error of length up to m errors, the generator polynomial G(x) must be of degree m. This is because the CRC algorithm is designed to detect burst errors that are shorter than the length of the CRC code.

When a burst error occurs, it can be represented as a polynomial E(x) of degree m or less. The received polynomial, denoted as R(x), is the sum of the transmitted polynomial T(x) and the error polynomial E(x). The received polynomial can be expressed as:

R(x) = T(x) + E(x)

When the received polynomial is divided by the generator polynomial G(x), a remainder polynomial is obtained. If there are no errors, the remainder polynomial will be zero. However, in the presence of errors, the remainder polynomial will be non-zero.

Since the generator polynomial G(x) is of degree m, the remainder polynomial will also have a degree of at most m-1. This means that the CRC algorithm can detect burst errors of up to m errors, as any burst error of length m or less will result in a non-zero remainder polynomial.

Therefore, the correct answer is option 'C' - m. The degree of the generator polynomial G(x) should be m in order to detect burst errors of up to m errors.