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Say the code has codewords, then there are codewords that differ from a codeword by a burst of length . Each of the M {\displaystyle M} words must be distinct, otherwise the code would have distance < 1 {\displaystyle <1} .
If the number of errors within a code word exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more uniform distribution of errors. [ 21 ]
Hence the rate of Hamming codes is R = k / n = 1 − r / (2 r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2 r − 1.
This table describes which parity bits cover which transmitted bits in the encoded word. For example, p 2 provides an even parity for bits 2, 3, 6, and 7. It also details which transmitted bit is covered by which parity bit by reading the column.
Given a prime number q and prime power q m with positive integers m and d such that d ≤ q m − 1, a primitive narrow-sense BCH code over the finite field (or Galois field) GF(q) with code length n = q m − 1 and distance at least d is constructed by the following method.
There are different encoding procedures for the Reed–Solomon code, and thus, there are different ways to describe the set of all codewords. In the original view of Reed & Solomon (1960), every codeword of the Reed–Solomon code is a sequence of function values of a polynomial of degree less than .
Toggle the table of contents. Code rate. 6 languages. ... implying that it is synonymous with net bit rate or useful bit rate exclusive of error-correction codes. ...
A code with distance d allows the receiver to detect up to transmission errors since changing positions of a codeword can never accidentally yield another codeword. Furthermore, if no more than ( d − 1 ) / 2 {\displaystyle (d-1)/2} transmission errors occur, the receiver can uniquely decode the received word to a codeword.
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