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Information Theory and Coding Basics

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?

  1. AAbout 0.43
  2. BAbout 0.57
  3. C1.75
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Answer: B. About 0.57

The code rate is k/n, the fraction of the codeword length n taken up by the k data bits. For the Hamming (7, 4) code it is 4/7 ≈ 0.57, so about 57% of the bits sent are data. A lower code rate means more check bits and makes it easier to resist errors, but lowers the efficiency of sending data.

  • A. About 0.43 is 3/7, the fraction of the codeword taken up by check bits.
  • B. Correct. 4/7 ≈ 0.57.
  • C. 1.75 is 7/4, with the numerator and denominator swapped. A code rate is 1 or less.
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More Information Theory and Coding Basics questions

  1. 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?
  2. 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?
  3. 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?
  4. Which of the following correctly describes ARQ (automatic repeat request) and FEC (forward error correction), methods for dealing with errors in communication?
  5. In a binary symmetric channel, 0s and 1s are sent and each bit flips independently with probability 0.1. With the binary entropy function H(p) = −p log₂ p − (1 − p) log₂(1 − p), H(0.1) ≈ 0.469. Which of the following is closest to the channel capacity of this channel per transmission?