Hamming Code Error Detection3/19/2021
Are you a researcher To avoid being denied access, log in if youre a ResearchGate member or create an account if youre not.No column consists of all zeros; each column is unique and has m elements.
![]() Hamming Code Error Detection Full Chapter URLView chapter Purchase book Read full chapter URL: Error-Control Coding Ali Grami, in Introduction to Digital Communications, 2016 10.4.3 Well-Known Codes There are many important linear block codes, including k 1, k single parity-check bit codes and CRC codes, which cannot correct errors and were discussed earlier as effective error-detection schemes. ![]() Repetition codes are the simplest type of linear block codes with error-correcting capability. A bit is encoded into a block of n identical bits, resulting in an ( n, 1) block code. Assuming the code has error-correcting capability t, a bit is encoded as a sequence of 2 t 1 identical bits, thus yielding 2 t 1, 1 linear block codes. In the case of hard-decision decoding, a majority logic decision needs to be made, i.e., if the number of 0s exceeds the number of 1s, the decoder decides in favor of a 0, otherwise, it decides in favor of a 1. The soft-decision decoding can be easily implemented to enhance the error rate performance at a modest level of complexity and the performance improvement due to soft-decision decoding can be significant. There are thus only two codewords in the code: all-zero codeword and all-one codeword. This code requires the use of significant bandwidth as it has a very low code rate 1 2 t 1 1 3, and therefore such codes are inefficient. However, repetition codes are attractive for deep-space communications, as there exists huge bandwidth at extremely high frequency band. Bose-Chaudhuri-Hocquenghem ( BCH) codes offer flexibility in the selection of the block length and code rate, and can be designed for correction of any given number of errors. A fast decoding algorithm can be employed for hard-decision decoding of the BCH codes. ![]() BCH codes can be defined in the binary field, such as the Hamming codes, and in the non-binary (symbol) field, such as the Reed-Solomon codes. For single error-correcting codes, if the total number of bits in a transmitted codeword is n, then m n k check bits must be able to indicate at least n 1 different states. Of these, one state means no error, and n states indicate the location of an error in each of the n positions, where it is also possible to have an error in the redundancy bits themselves. So n 1 states must be discoverable by n k bits, and n k bits can indicate 2 n k different states. Therefore, we must have 2 n k n 1 or equivalently 2 m 1 n, for an ( n, k ) code with single error-correcting capability. Hamming codes have d min 3, and thus t 1, i.e., a single error can be corrected regardless of the number of parity-check bits. An ( n, k ) Hamming code has m n k parity-check bits, where n 2 m 1 and k 2 m 1 m, for m 3. The parity-check matrix H of a Hamming code has m rows and n columns, and the last n k columns must be chosen such that it forms an identity matrix.
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |