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Burst error-correcting code. In coding theory, burst error-correcting codes employ methods of correcting burst errors, which are errors that occur in many consecutive bits rather than occurring in bits independently of each other. Many codes have been designed to correct random errors.
They also include brief discussions of additional material not covered in more detail later, including information theory, convolutional codes, and burst error-correcting codes. Chapter 3 presents the BCH code over the field G F ( 2 4 ) {\displaystyle GF(2^{4})} , and Chapter 4 develops the theory of finite fields more generally.
The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors.
A fire code can correct all burst errors of length t or less if no two bursts () and ′ appear in the same co-set. This can be proved by contradiction. This can be proved by contradiction. Suppose there are two distinct nonzero bursts b ( x ) {\displaystyle b(x)} and x j b ′ ( x ) {\displaystyle x^{j}b'(x)} of length t {\displaystyle t} or ...
Viterbi decoder. A Viterbi decoder uses the Viterbi algorithm for decoding a bitstream that has been encoded using a convolutional code or trellis code . There are other algorithms for decoding a convolutionally encoded stream (for example, the Fano algorithm ). The Viterbi algorithm is the most resource-consuming, but it does the maximum ...
The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF (2) (the integers modulo 2 ), that is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the coefficients of a message polynomial of this ...
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. Therefore, interleaving is widely used for burst error-correction.
The BCH code with and higher has the generator polynomial. This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial repetition code ).