It is simple to find the minimum Hamming distance for a linear block code. The scheme in Table above is also a linear block code.
For example, the XORing of the second and With any other codeword is a valid codeword. The scheme in Table above is a linear block code because the result of XORing any codeword OR (addition modulo-2) of two valid codewords creates another valid codeword. Hamming distance in a block code must be dmin = 2t + 1.Ī linear block code is a code in which the exclusive To guarantee correction of up to t errors in an cases, the minimum Error correctionĬodes need to have an odd minimum distance (3, 5, 7. In other words, if this code is used for error correction, part of its capability is wasted. This code guarantees the detection of up to three errors (s = 3), but it can correct up to one error.
What is the error detection and correction A code scheme has a Hamming distance dmin = 4. Received codeword and the errors are undetected. Of three errors change a valid codeword to another valid codeword. Our second block code scheme (Table below) has dmin = 3.Īgain, we see that when any of the valid codewords is sent, two errors create a codeword which The received codeword may match a valid codeword and the errors are not detected. Occurs, the received codeword does not match any valid codeword. This code guaranteesįor example, if the third codeword (101) is sent and one error The minimum Hamming distance for our first code scheme (Table given below) is 2. The distances are not enough (s + 1) for the receiver to accept it as valid. In other words, if the minimum distance between all validĬodewords is s + 1, the received codeword cannot be erroneously mistaken for anotherĬodeword. If our code is to detect up to "s" errors, the minimumĭistance between the valid codes must be "s + 1", so that the received codeword does Sent codeword and received codeword is "s". If "s" errors occur during transmission, the Hamming distance between the In other words, the Hamming distanceīetween the received codeword and the sent codeword is the number of bits that are corruptedįor example, if the codeword 00000 is sent and 01101 is received, 3 bitsĪre in error and the Hamming distance between the two is d(00000, 01101) = 3. Is the number of bits affected by the error. Is corrupted during transmission, the Hamming distance between the sent and received codewords Smallest Hamming distance between all possible pairs. In a set of words, the minimum Hamming distance is the Note that the Hamming distance isĮxample :The Hamming distance d(000, 011) is 2 because 000 ⊕ 011 is 011 (two 1's) Two words and count the number of 1's in the result. The Hamming distance can easily be found if we apply the XOR operation (⊕)on the The Hamming distance between two words (of the same size) is the number ofĭifferences between the corresponding bits.
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