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
Erasure Correcting,
Error Detecting, and
Error Correcting Codes
1000101001001000100100010
1000
1000101001001000100100010
10001
Most significant bit is 1:
XOR 11011
Drop leading 0
01010
1010
1000101001001000100100010
Shift contents of register to left, drop in next bit
1010
0
Most significant bit is 0:
Drop leading 0
0100
1000101001001000100100010
Shift contents of register to left, drop in next bit
0100
1
Most significant bit is 0:
Drop leading 0
1001
1000101001001000100100010
Shift contents of register to left, drop in next bit
10100
Most significant bit is 1:
XOR 11011
Drop leading 0
01111
1111
Decoding Table
Maximum Likelihood Decoding
00 - 00100
01 - 01110
10 - 10001
11 - 11000
11111
How do we decode?
Hamming matrix for r = 3.
(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).
(0100010)· H = (111).