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Proof. We need to prove that if you add a burst of length to a codeword (i.e. to a polynomial that is divisible by ()), then the result is not going to be a codeword (i.e. the corresponding polynomial is not divisible by ()).
A checksum of a message is a modular arithmetic sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a ones'-complement operation prior to transmission to detect unintentional all-zero messages.
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.
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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 ]
Deep-space concatenated coding system. [8] Notation: RS(255, 223) + CC ("constraint length" = 7, code rate = 1/2). One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the Voyager program.
Some 1s bit in the information part of the code word have changed to 0s. Some 1s bits in the check (or redundant) portion of the code word have changed to 0s. Some 1s bits in both the information and check portions have changed to 0s. For case 1, the number of 0-valued bits in the information section will, by definition of the error, increase.
The rate of a block code is defined as the ratio between its message length and its block length: = /. A large rate means that the amount of actual message per transmitted block is high.