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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 ]
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} .
Ignoring any lines going out of the picture, there are eight possible six-bit strings corresponding to valid codewords: (i.e., 000000, 011001, 110010, 101011, 111100, 100101, 001110, 010111). This LDPC code fragment represents a three-bit message encoded as six bits. Redundancy is used, here, to increase the chance of recovering from channel ...
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.
The rate of a code is inversely related to the query complexity, but the exact shape of this tradeoff is a major open problem. [8] [9] It is known that there are no LDCs that query the codeword in only one position, and that the optimal codeword size for query complexity 2 is exponential in the size of the original message. [8]
A Reed–Solomon code (like any MDS code) is able to correct twice as many erasures as errors, and any combination of errors and erasures can be corrected as long as the relation 2E + S ≤ n − k is satisfied, where is the number of errors and is the number of erasures in the block.
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.