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COEN 180

COEN 180. Erasure Correcting, Error Detecting, and Error Correcting Codes. Basics. Use encoding to protect data for storage for transmission. Encode in order to Discover errors Example: CRC code tagged onto IP packets. Example: ISBN number check digit Correct errors

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COEN 180

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  1. COEN 180 Erasure Correcting, Error Detecting, and Error Correcting Codes

  2. Basics • Use encoding to protect data • for storage • for transmission. • Encode in order to • Discover errors • Example: CRC code tagged onto IP packets. • Example: ISBN number check digit • Correct errors • Example: Disk Drives, Communications Protocols • Correct erasures • Redundant storage of data (e.g. in Disk Arrays)

  3. Basics • Rich Mathematical Theory • Block codes, burst codes, … • Uses heavily Galois fields • Mathematical structure with addition, multiplication, … in which we manipulate objects according to the same rules as the more popular fields of real numbers, complex numbers, rational numbers,

  4. Error Detection • Basic Principle: • Calculate a small “signature” from a data object and store it with the data object. • For verification, recalculate the signature. • Example: CRC codes • Cyclic Redundancy Codes • Fast calculation

  5. Error Detection • Parity code • Block code of l bits • Adds parity bit to block code. • Example: • Message: • 1011 1000 0100 • Encoded message: • 1011 1000 0100 1 • Encoded message after one error: • 1111 1000 0100 1  Parity is Off • Encoded message after two errors: • 1111 1100 0100 1  Parity is correct, error undetected

  6. Error Detection: CRC • Example: • Need a CRC polynomial (a bit string for us) • Example 11011 (with leading 1) • Cyclically shift through the message with a field the length of the CRC polynomial • Example 1000101001001000100100010 • Whenever most significant bit is 1, XOR CRC polynomial.

  7. Error Detection: CRC • Example: 1000101001001000100100010 1000

  8. Error Detection: CRC • Example: 1000101001001000100100010 10001 Most significant bit is 1: XOR 11011 Drop leading 0 01010 1010

  9. Error Detection: CRC • Example: 1000101001001000100100010 Shift contents of register to left, drop in next bit 1010 0 Most significant bit is 0: Drop leading 0 0100

  10. Error Detection: CRC • Example: 1000101001001000100100010 Shift contents of register to left, drop in next bit 0100 1 Most significant bit is 0: Drop leading 0 1001

  11. Error Detection: CRC • Example: 1000101001001000100100010 Shift contents of register to left, drop in next bit 10100 Most significant bit is 1: XOR 11011 Drop leading 0 01111 1111

  12. Error Detection: CRC • Add CRC of packet in TCP/IP

  13. Error Correction • Transmission Channel • Used for communications channel • Used for storage channel (travel in time)

  14. Error Correction • Encode an object with additional parity bits. • Use parity bits to detect and correct an error. • Trivial Example: Replication Code • Encodes messages of length 1b • Add to bit two copies of the bit. • When all three bits in the received message coincide, conclude that no error has occurred. • Otherwise, use a majority rule to decide. Decoding Table

  15. Error Correction • Hamming Distance • Defined to be the number of different bits in a bit string of length l. • Examples: • 0011 1110 and 0101 1110 have distance 2 • 0011 1110 and 1100 0001 have distance 8

  16. Error Correction • Code Word: a word that can be obtained by encoding a piece of datum. • There are many fewer code words than possible. • Maximum Likelihood Decoding: • When receiving a message, decode it with the code word closest in Hamming distance.

  17. Error Correction Maximum Likelihood Decoding

  18. Error Correction • Example: • Assume the following (bad) code: 00  -    00100 01  -    01110 10  -    10001 11  -    11000 • Assume we received 11111 How do we decode?

  19. Error Correction • Linear Codes • Based on Reed Solomon. • Encode symbols: bit-strings of length l. • Generate an m by n matrix G such that • any m by m submatrix is invertible. • such that G is of the form (1|P) • where 1 is the m by m identity matrix • P is the so-called parity matrix. • Encode m symbol data (x1, x2, …,xm) as (x1,x2,…,xm)G

  20. Error Correction : Hamming Code • Hamming Code • Systematic: Code word is data word + parity • m message bits and r parity bits • Linear • Create matrix of size 2r-1by r. • Populate rows with binary encoding of all numbers from 1 to 2r • If desired, arrange them so that the identity matrix is the lower part of the matrix.

  21. Error Correction : Hamming Code Hamming matrix for r = 3.

  22. Error Correction: Hamming Code • If data word is (a,b,c,d), then code word is (a,b,c,d,x,y,z) where the parity bits are chosen such that (a,b,c,d)H=(0,0,0,0,0,0,0).

  23. Error Correction: Hamming Code

  24. Error Correction: Hamming Code • Example: • Data is (0,1,0,1) • What is the code word?

  25. Error Correction: Hamming Code • Answer: • (0,1,0,1,0,1,0).

  26. Error Correction: Hamming Code • To correct an error, apply H to the received vector. • The result is the binary encoding of a row in H. • The corresponding coefficient in the vector is were the error most likely was.

  27. Error Correction: Hamming Code • Assume that we received (0100010). • We apply H to this vector: • Therefore, the error is given by row (111). • That is, in the fourth row. • Hence, error is in bit position 4. (0100010)· H = (111).

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