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It is not difficult to show that only single bit errors can be corrected using this scheme. Alternatively, Hamming codes can be used to detect single and double bit errors, by merely noting that the product of H is nonzero whenever errors have occurred. In the adjacent diagram, bits 4 and 5 were flipped.
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.
The main idea is to choose the error-correcting bits such that the index-XOR (the XOR of all the bit positions containing a 1) is 0. We use positions 1, 10, 100, etc. (in binary) as the error-correcting bits, which guarantees it is possible to set the error-correcting bits so that the index-XOR of the whole message is 0.
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In fact, the term "list-decoding capacity" should actually be read as the capacity of an adversarial channel under list decoding. Also, the proof for list-decoding capacity is an important result that pin points the optimal trade-off between rate of a code and the fraction of errors that can be corrected under list decoding.
The block length of a block code is the number of symbols in a block. Hence, the elements of are strings of length and correspond to blocks that may be received by the receiver.
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
The normal deviate mapping (or normal quantile function, or inverse normal cumulative distribution) is given by the probit function, so that the horizontal axis is x = probit(P fa) and the vertical is y = probit(P fr), where P fa and P fr are the false-accept and false-reject rates.