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Information Theory and Coding Basics

A single even parity bit is added to a bit string of data so that the number of 1s is even, and the receiver checks the number of 1s. Which of the following correctly describes this check?

  1. AIt detects a 1-bit error and also shows which bit is wrong, so the receiver can correct it
  2. BIt detects an error when 2 bits are wrong at the same time, just as it does for a 1-bit error
  3. CIt detects an odd number of bit errors but cannot detect an even number of bit errors
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Answer: C. It detects an odd number of bit errors but cannot detect an even number of bit errors

If one bit flips, the parity (oddness or evenness) of the number of 1s changes, so the check reveals the error. If two bits flip, the parity returns to what it was, so the error goes unnoticed. In general, an odd number of errors is detected and an even number is not. Moreover, the check does not show which bit is wrong, so it cannot correct the error, and a response such as asking for retransmission is needed.

  • A. It shows that there is an error, but not which bit. Correction needs a Hamming code or similar.
  • B. When two bits flip, the parity of the number of 1s returns to what it was, so the error is not detected.
  • C. Correct. The parity of the number of 1s changes only when an odd number of bits are wrong.
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More Information Theory and Coding Basics questions

  1. Four rows of 4-bit data were sent with an even parity bit for each row at its right end and a final row of even parity bits for each column (vertical and horizontal parity). The received bits were row 1: 1011 1, row 2: 0100 0, row 3: 1100 0, row 4: 0101 0 and parity row: 0100 (the rightmost bit of each row is its row parity). If exactly one bit is wrong, which of the following is its position?
  2. 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?
  3. 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?
  4. Which of the following correctly describes a CRC whose generator polynomial has degree r (r check bits) and a constant term of 1?
  5. Which of the following is the Hamming distance between the two 7-bit codewords 1011010 and 0110011?