English
Menu

Information Theory and Coding Basics

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?

  1. AMinimum Hamming distance 4; corrects up to 1 bit
  2. BMinimum Hamming distance 3; corrects up to 2 bits
  3. CMinimum Hamming distance 3; corrects up to 1 bit
Show answer and explanation

Answer: C. Minimum Hamming distance 3; corrects up to 1 bit

Of the six pairs, 00000 and 11110, and 01011 and 10101, are at Hamming distance 4, and the other four pairs are at distance 3, so the minimum Hamming distance is 3. It can correct up to (3 − 1) ÷ 2 = 1 bit. This code is a linear code, in which the exclusive OR of any two codewords is also a codeword, so the value 3 can also be found as the smallest number of 1s in a nonzero codeword.

  • A. 4 is the largest distance. The number of errors that can be corrected is set by the distance between the two nearest codewords, that is, the minimum.
  • B. 2 is d − 1, the value for detection only. Correction works only up to (d − 1) ÷ 2 rounded down.
  • C. Correct. The minimum Hamming distance is 3, so errors of up to 1 bit can be corrected.
Practice this topic Back to topic

Books on Information Theory and Coding Basics Ad

As an Amazon Associate I earn from qualifying purchases.

More Information Theory and Coding Basics questions

  1. 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?
  2. 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?
  3. 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?
  4. 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?
  5. Which of the following correctly describes ARQ (automatic repeat request) and FEC (forward error correction), methods for dealing with errors in communication?