![]() ![]() ![]() If bit "Y" is set in the received word, then the recomputed check bits will be "000", and the syndrome will be "010", which is the bit position for Y. If bit "B" is set in the received word, then the recomputed check bits X'Y'Z' (and the syndrome) will be 110, which is the bit position for B. Let's assume that the data bits are all zero, which also means that all of the check bits are zero as well. If the bit in this position is flipped, then the original 7-bit codeword is perfectly reconstructed.Ī couple of examples will illustrate this. But if there's a single bit error in any of the seven received bits, the result of the XOR is a nonzero three-bit number called the "syndrome" that directly indicates the position of the bit error as defined in the table above. If there's no error in the received bits, the result of the XOR is all zeros. If the new check bits are XOR'd with the received check bits, an interesting thing occurs. When they're received (or retrieved) later, the data bits are put through the same encoding process as before, producing three new check bits X', Y' and Z'. ![]() Now all seven bits - the codeword - are transmitted (or stored), usually reordered so that the data bits appear in their original sequence: A B C D X Y Z. Similarly, the check bit Y is the parity bit for all of the bits with a "1" in the second row (A, B and D), and the check bit Z is the parity bit for all of the bits with a "1" in the third row (A, C and D). ![]() The check bit X is set or cleared so that all of the bits with a "1" in the top row - A, B C and X - have even parity. If the four data bits are called A, B, C and D, and our three check bits are X, Y and Z, we place them in the columns such that the check bits are in the columns with one "1" and the data bits are in the columns with more than one "1". If you number the bit positions of an 8-bit word in binary, you see that there is one position that has no "1"s in its column, three positions that have a single "1" each, and four positions that have two or more "1"s. It takes three check bits to protect four data bits (the reason for this will become apparent shortly), giving a total of 7 bits in the encoded word. Each check bit is a parity bit for a particular subset of the data bits, and they're arranged so that the pattern of parity errors directly indicates the position of the bit error. Hamming codes are relatively easy to construct because they're based on parity logic. This type of error recovery is also known as forward error correction (FEC).Ĭonstructing a Hamming code to protect, say, a 4-bit data word Such codes are used in data transmission or data storage systems in which it is not feasible to use retry mechanisms to recover the data when errors are detected. A Hamming code is a particular kind of error-correcting code (ECC) that allows single-bit errors in code words to be corrected. ![]()
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