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Information Theory and Coding Basics

The minimum Hamming distance of a code (the smallest Hamming distance between any two codewords) is 5. Consider using it only for error detection, or only for error correction. Which of the following correctly pairs the number of bit errors that can always be detected with the number that can always be corrected?

  1. ADetection up to 4 bits, correction up to 2 bits
  2. BDetection up to 5 bits, correction up to 2 bits
  3. CDetection up to 4 bits, correction up to 4 bits
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Answer: A. Detection up to 4 bits, correction up to 2 bits

In a code with minimum Hamming distance d, an error of up to d − 1 bits never turns a codeword into another codeword, so it can be detected. Correction changes the received bit string to the nearest codeword, so if the number of errors is at most (d − 1) ÷ 2 rounded down, the original codeword remains the nearest. For d = 5, detection works up to 4 bits and correction up to 2 bits.

  • A. Correct. Detection up to d − 1 = 4 bits and correction up to (5 − 1) ÷ 2 = 2 bits.
  • B. A 5-bit error can turn a codeword exactly into another codeword, in which case it is not detected.
  • C. With 3 or more bit errors, another codeword can be nearer than the original, and the word is corrected to the wrong codeword.
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More Information Theory and Coding Basics questions

  1. Which of the following correctly pairs the minimum Hamming distance of the code consisting of the four codewords 00000, 01011, 10101 and 11110 with the number of bit errors it can always correct?
  2. Which of the following is closest to the code rate of the Hamming (7, 4) code, which adds 3 check bits to 4 bits of data to form a 7-bit codeword?
  3. In a Hamming (7, 4) code, the seven bits are numbered 1 to 7 from the left, and bits 1, 2 and 4 are check bits. They are set so that the number of 1s is even in bits 1, 3, 5 and 7 (check 1), in bits 2, 3, 6 and 7 (check 2) and in bits 4, 5, 6 and 7 (check 4). If the received word 0110001 has a single-bit error, which of the following is the wrong bit?
  4. A SECDED code, which corrects 1-bit errors and detects 2-bit errors, is added to 32 bits of data. The number of check bits r of a Hamming code is the smallest value satisfying 2^r ≥ number of data bits + r + 1, and SECDED adds 1 bit of overall parity to this. Which of the following is the number of check bits needed?
  5. Five Hamming (7, 4) codewords are lined up and sent in the order of bit 1 of each of the five codewords, then bit 2 of each, and so on (interleaving of depth 5). The receiver restores the original order and then corrects a 1-bit error in each codeword. Which of the following is the maximum length of a burst error, concentrated in consecutive bits, that can always be corrected?