The Hamming code uses the parity-check matrix used earlier in this chapter
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Quantum Information and Technology, Ch Quantum error correction
Problem
The Hamming [7,4,3] code uses the parity-check matrix M used earlier in this chapter, whose column j is the integer j written in binary, most significant bit on top.
((a)) A single bit-flip error strikes position 5 of a transmitted codeword. Give the syndrome Me the decoder measures, and say which bit it flips to correct it.
((b)) Suppose instead that bit-flip errors strike both position 1 and position 4 at once. Give the syndrome this error pattern produces, and say what the decoder does with it.
((c)) State the total number of bits that differ from the originally sent codeword once the decoder has acted in case (b), and say whether the codeword is recovered correctly.