1. How many errors may be detected (not necessarily corrected) if a code has a Hamming Distance of 6?
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1 Answers to Practice Problems Practice Problems - Hamming distance 1. How many errors may be detected (not necessarily corrected) if a code has a Hamming Distance of 6? 2n = 6; n=3 2. How many errors may be corrected if a code has a Hamming Distance of 9? 2n+1 = 9; n=4 3. How many errors may be accurately detected for a Hamming Distance of m where m is even? How many may be corrected? Detected: 2n = m; n = m/2 Corrected: n =m/2-4. How many errors may be accurately detected for a Hamming Distance of m, where m is odd? How many may be corrected? Detected: Corrected: 2n = m = (m-1)+1 (where m-1 is even); n=(m-1)/2 n = (m-1)/2 5. If a code can detect (but not correct) up to 5 errors, what is the Hamming Distance for that code? 2 n = 2 x 5 = If a code can correct at most 4 errors, what is the Hamming Distance of that code? 2n + 1 = 2 x = 9 Practice Problems - Parity 1. Assuming odd-parity codewords, what is the value of the parity bit ( 1 or 0) which should be added to each of the following data words? a Parity bit = 1 Codeword = NTC 11/15/04 136
2 b Parity bit = 0 Codeword = c Parity bit = 1 Codeword = d Parity bit = 0 Codeword = e Parity bit = 0 Codeword = f. 0 Parity bit = 1 Codeword = 01 g Parity bit = 1 Codeword = h. 11 Parity bit = 1 Codeword = Assuming even-parity codewords, which of the following codewords have errors? a No Error c Error b Error d Error 3. If two 9-bit codewords are concatenated to form a new 18-bit codeword (containing all the data and both parity bits of the original codewords), what is the parity of the new codeword? Even. 4. If a parity error checker determines that there is, indeed, an error in a codeword, how many errors are in the word? An odd number of errors may have occurred 5. If an even number of errors occur in a codeword, how many are detected by a parity error checker? None. Practice Problems - Checksum 1. What is the checksum for each of the following additions using 2's complement addition? a. 101 b Repeat Exercise 1. using 1's complement arithmetic Find the 4-bit 1's complement checksum for the data 2's compl sum = NTC 11/15/04 137
3 Assuming a 4-bit 2's complement checksum, are there errors in the following codewords? a yes b no 5. Using 1's complement arithmetic (and complementing the result) what is the checksum value for the following data stream, assuming a 5-bit checksum? Practice Problems - CRC Codes 1. Which of the following polynomials is a valid generator polynomial? a. X+1 Y c. X 2 +X+1 N e. X 9 +X 7 +X 3 +X 2 +X+1 Y b. X 2 +1 Y d. X 10 +X 8 +X N f. X 3 +X 2 +1 N 2. What is the largest burst error size guaranteed to be detected by each of the following generator polynomials a. x c. X 12 +X 11 +X 3 +X+1 12 b. X 16 +X 15 +X d. X 16 +X 12 +X Using the generator polynomial X 7 +X 6 +X 4 +X 2 +X+1, find the CRC bits for the following data words. CRC= CRC= a b Using the generator polynomial X 4 +X 3 +X+1, determine which of the following received codewords has errors. CRC =1010 CRC=0100 a b Practice Problems - Number of Hamming Check Bits 1. What is the minimum number of check bits required for single-error correcting codes with the following numbers of data bits? a. 1 Ans: 2 d. 57 Ans: 6 b. 33 Ans: 6. e. 11 Ans: 4 NTC 11/15/04 138
4 c. 3 Ans: 3 f. 64 Ans: 7 2. Which of the codes generated above is a perfect code? a., d. and e. 3. What is the maximum number of data bits possible in codes with each of the following number of check bits? a. 1 Ans: 0 c. 5. Ans: 26 b. 2 Ans: 1 d. 6 Ans: How many check bits are there in each of the following codes, assuming codewords of the given lengths, and assuming a minimum number of check bits? a. 8 Ans: 4 c. 32 Ans: 6 b. 12 Ans: 4 d. 128 Ans: 8 Practice Problems - Determining Syndromes Consider the following H-Matrix d 0 d 1 d 2 d 3 d 4 c 0 c 1 c 2 c What are the equations which determine c 0, c 1, c 2, and c 3 (refer to [E4])? c 0 = d 0 d 1 c 1 = d 0 d 2 d 3 c 2 = d 1 d 2 d 4 c 3 = d 3 d 4 2. Given the following data words, find the corresponding codewords a Ans: c Ans: b Ans: d Ans: Given the following codewords, determine which have errors, and the error location. For those with errors, determine the original codeword. NTC 11/15/04 139
5 a No error b Error in bit d0, original codeword = c Error in bit c1, original codeword = Practice Problems - Developing H-matrices 1. Develop H-Matrices for codes with the following numbers of data bits a. 1 b. 2 c. 7 a. d 0 c 0 c 1 b. d 0 d 1 c 0 c 1 c 2 c. d 0 d 1 d 2 d 3 d 4 d 5 d 6 c 0 c 1 c 2 c How many rows will there be in an H-Matix, assuming a. There are 5 check bits Ans: 5 b. There are 64 data bits Ans: 7 Practice Problems - SEC-DED codes Consider the following H-Matrix (from an earlier Practice Problem set.) d 0 d 1 d 2 d 3 d 4 c 0 c 1 c 2 c Modify the matrix so that it provides double error detection as well as single error correction. d 0 d 1 d 2 d 3 d 4 c 0 c 1 c 2 c 3 c NTC 11/15/04 140
6 2. What is the equation which determines c 4? c 4 = d 0 d 1 d 2 d 3 d 4 c 0 c 1 c 2 c 3 3. Given the following data words, find the corresponding codewords a Ans: c Ans: b Ans: d Ans: Given the following codewords, determine which have errors, and whether the errors are single errors or double errors a Double Error s = 01010; non-zero, s 4 = 0 b Single Error s = 00101; non-zero, s 4 = 1 c Double Error s = 01010; non-zero, s 4 = 0 d No Error s = 00000; zero NTC 11/15/04 141
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