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Information Theory and Coding Basics

Data are added up one byte at a time, and the low 8 bits of the sum are sent along as a check value (checksum). Which of the following errors cannot be detected by this check?

  1. AAn error in which just one bit flips within one of the bytes
  2. BAn error in which the value of one of the bytes becomes larger by 1
  3. CAn error in which the order of two bytes is swapped
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Answer: C. An error in which the order of two bytes is swapped

Addition gives the same result in any order, so if two bytes are swapped the sum stays the same and the error cannot be detected. A single bit flip, or a change of 1 in the value of one byte, always changes the low 8 bits of the sum, so it is detected. With a CRC, the effect on the remainder of the division depends on the position of each bit, so in many cases such swaps are also detected.

  • A. A single bit flip changes the sum by 2^k (k from 0 to 7), so the low 8 bits always change and the error is detected.
  • B. The sum changes by 1, so the low 8 bits change too and the error is detected.
  • C. Correct. Changing the order of addition leaves the sum the same, so the error is not detected.
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More Information Theory and Coding Basics questions

  1. Three CRC check bits for the generator polynomial x³ + x + 1 (bit string 1011) are to be appended to the data 1101. The check bits are the remainder of dividing 1101000, the data followed by three 0s, by 1011, using exclusive OR instead of subtraction in the division. Which of the following are the check bits?
  2. Which of the following correctly describes a CRC whose generator polynomial has degree r (r check bits) and a constant term of 1?
  3. Which of the following is the Hamming distance between the two 7-bit codewords 1011010 and 0110011?
  4. 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?
  5. 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?