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Clearly, all codewords are distinct. If we puncture the code by deleting the first letters of each codeword, then all resulting codewords must still be pairwise different, since all of the original codewords in have Hamming distance at least from each other. Thus the size of the altered code is the same as the original code.
Linearity guarantees that the minimum Hamming distance d between a codeword c 0 and any of the other codewords c ≠ c 0 is independent of c 0. This follows from the property that the difference c − c 0 of two codewords in C is also a codeword (i.e., an element of the subspace C), and the property that d(c, c 0) = d(c − c 0, 0). These ...
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A code is considered "binary" if the codewords use symbols from the binary alphabet {,}. In particular, if all codewords have a fixed length n , then the binary code has length n . Equivalently, in this case the codewords can be considered elements of vector space F 2 n {\displaystyle \mathbb {F} _{2}^{n}} over the finite field F 2 ...
Request that the codeword be resent – automatic repeat-request. Choose any random codeword from the set of most likely codewords which is nearer to that. If another code follows, mark the ambiguous bits of the codeword as erasures and hope that the outer code disambiguates them; Report a decoding failure to the system
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that must be satisfied for the vector (,,,) to be a codeword of C. From the definition of the parity-check matrix it directly follows the minimum distance of the code is the minimum number d such that every d - 1 columns of a parity-check matrix H are linearly independent while there exist d columns of H that are linearly dependent.