Transcript COEN 180 Erasure Correcting, Error Detecting, and Error Correcting Codes

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



Correct errors


Example: CRC code tagged onto IP packets.
Example: ISBN number check digit
Example: Disk Drives, Communications Protocols
Correct erasures

Redundant storage of data (e.g. in Disk Arrays)
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,
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
Error Detection

Parity code
 Block code of l bits
 Adds parity bit to block
 Example:
 Message:


1011 1000 0100 1
Encoded message after one error:


1011 1000 0100
Encoded message:


code.
1111 1000 0100 1  Parity is Off
Encoded message after two errors:

1111 1100 0100 1  Parity is correct, error undetected
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.

Error Detection: CRC

Example:
1000101001001000100100010
1000
Error Detection: CRC

Example:
1000101001001000100100010
10001
Most significant bit is 1:
XOR 11011
Drop leading 0
01010
1010
Error Detection: CRC

Example:
1000101001001000100100010
1010 0
Shift contents of
register to left, drop in
next bit
Most significant bit is 0:
Drop leading 0
0100
Error Detection: CRC

Example:
1000101001001000100100010
0100 1
Shift contents of
register to left, drop in
next bit
Most significant bit is 0:
Drop leading 0
1001
Error Detection: CRC

Example:
1000101001001000100100010
10100
Shift contents of
register to left, drop in
next bit
Most significant bit is 1:
XOR 11011
Drop leading 0
01111
1111
Error Detection: CRC

Add CRC of packet in TCP/IP
Error Correction

Transmission Channel
 Used
for communications channel
 Used for storage channel (travel in time)
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
000
0
100
0
001
0
101
1
010
0
110
1
011
1
111
1
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

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.
Error Correction
Maximum Likelihood Decoding
Error Correction

Example:
 Assume
00
01
10
11
-
the following (bad) code:
00100
01110
10001
11000
 Assume
we received
11111
How do we decode?
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.
m symbol data (x1, x2, …,xm) as
(x1,x2,…,xm)G
 Encode
Error Correction : Hamming Code

Hamming Code
 Systematic:

Code word is data word + parity
m message bits and r parity bits
 Linear
matrix of size 2r-1 by 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.
 Create
Error Correction : Hamming Code
Hamming matrix for r = 3.
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).

Error Correction: Hamming Code
Error Correction: Hamming Code

Example:
 Data
is (0,1,0,1)
 What is the code word?
Error Correction: Hamming Code

Answer:
 (0,1,0,1,0,1,0).
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.

Error Correction: Hamming Code


Assume that we received (0100010).
We apply H to this vector:
(0100010)· H = (111).



Therefore, the error is given by row
(111).
That is, in the fourth row.
Hence, error is in bit position 4.