The remainder after division by x 4 + x + 1 is x 3 + x 2 + x (CRC bits:1110), which is nonzero. The errors are detectedĬonclusion: The string 000110110111100 is received, corresponding to x 11 + x 10 + x 8 + x 7 + x 5 + x 4 + x 3 + x 2. The remainder after division by x 4 + x + 1 (10011) is x 3 + x 2 + x (1110), which is nonzero. Quotient / Result is: x 7 + x 6 + x 3 + x 2 + x The errors are detectedĭividend (Polynomial): x 11 + x 10 + x 8 + x 7 + x 5 + x 4 + x 3 + x 2 The remainder after division by 10011 is 1110, which is nonzero. What is received? Can the error be detected?ĬRC polynomial as binary sequence: Data Error Detect binary sequence: Suppose the channel introduces an error pattern 100010000000000 (i.e., a flip from 1 to 0 or from 0 to 1 in position 1 and 5) from the data in Question 1.

Quotient / Result is: x 10 + x 6 + x 4 + x 2 Therefore, we append 4 zeros at the end of the data/message.ĭividend (Polynomial): x 10 + x 7 + x 4 + x 3 + x + 1 The red zeroes '0' are the bit sequence to which append a number N of 0’s, where N=The length of CRC polynomial minus 1.Į.g.: CRC polynomial '10011' have the length of 5 bits, N=5 - 1=4 bits. <- (1001=1 (1 bit)(*XOR mean minus each bit individually with unsigned result), and move the next 4 bit to fit the bit length of the CRC polynomial, which is 1 with 0110=10110, the divisor moves over to align with the next 1 in the dividend) <- (input right padded by 4 bits (bit length of the CRC polynomial minus one 5-1=4)) <- the quotient result is different than the actual division, which using binary subtraction. Show the steps clearly and derive the solution.Īnswer: CRC polynomial as binary sequence: Message as binary sequence: Given a CRC generator x 4 + x + 1 (10011), calculate the CRC code for the message 10010011011. In a CRC error-detecting scheme, choose the generator polynomial as x 4 + x + 1 (10011). This tools will display the answer in form, which is Excel friendly, and as plain text for better compatibility in text-only environment.

The XOR and addition operation can always be performed bitwise-parallel.īack in the days when question like this is kinda time-consuming, here is a calculator which generate the step-by-step solution for all these problems. However, binary long division uses binary subtraction and “CRC long division” uses XOR operation. The polynomial coefficients are calculated according to the finite-field arithmetic as the binary long division. message binary is right padded with the bit length minus one of the CRC polynomial minus one -1) as the “dividend”, and the “remainder” becomes the result of CRC bits. The CRC code requires definition of a so-called “generator polynomial” as the “divisor”.Īnd takes the message binary-shifted to left and extend the bit length with the divisor bit length minus one -1 (a.k.a. CRC is an error-detecting code is based on binary / polynomial “division”, and the sequence of redundant bits is appended to the end of a data unit so that the resulting data unit becomes exactly divisible (remainder=0) by a second predetermined binary number.

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