1. MEGGIT DECODER

2. PROBLEM STATEMENT To correct all single errors in n-bit codeword, n error patterns of single errors and their corresponding syndromes need to be stored (Complexity ) ES 1 0 ---- 0 0 1 ---- 0 0 0 ---- 1 n error patterns of single error

3. SOLUTION Store the syndrome of the error pattern that corresponds to an error in the last bit ONLY. Cyclically shift the erroneous codeword till the error becomes in the last bit The syndrome of the shifted codeword matches the stored one Correct the shifted Codeword Cyclically shift the corrected codeword till the corrected bit returns back to its original location

4. EXAMPLE Consider a (7,4) Cyclic Code with g(X)=1+X+X 3 Encoder Decoder u(X)=1+X+X 2 +X 3 v(X)= 1+X 3 +X 5 +X 6 e(X)= X 2 r(X)= 1+X 2 +X 3 +X 5 +X 6 u= 1 1 1 1 v= 1 0 0 1 0 1 1 e= 0 0 1 0 0 0 0 r= 1 0 1 1 0 1 1 ES 1 0 0 0 0 0 01 0 0 0 1 0 0 0 0 00 1 0 0 0 1 0 0 0 00 0 1 0 0 0 1 0 0 01 1 0 0 0 0 0 1 0 00 1 1 0 0 0 0 0 1 01 1 1 0 0 0 0 0 0 11 0 1 E=0 0 0 0 0 0 1S=1 0 1

5. r= 1 0 1 1 0 1 1 Syndrome Circuit S=101? r (1) = 1 1 0 1 1 0 1 No r (2) = 1 1 1 0 1 1 0r (3) = 0 1 1 1 0 1 1r (4) = 1 0 1 1 1 0 1 Correct Yes r= 1 0 1 1 0 1 1

6. SYNDROME CIRCUIT Syndrome Circuit of (7,4) Cyclic Code with g(X)=1+X+X 3 n=7, k=4 n-k=3 shift registers r=0 0 1 0 1 1 0 r(X)= X 2 +X 4 +X 5 000 0 0 000 00 0 0 0 1 000 0 0 1 0 0 1 001 0 0 1 1 0 0 110 0 0 1 1 0 1 011 1 1 0 1 1 0 011 1 1 1 1 1 111 1 1 1 0 1 s=1 0 1

7. r (1) = 0 0 0 1 0 1 1 r (1) (X)= X 3 +X 5 +X 6 000 1 0 000 00 100 1 100 00 110 0 011 00 011 1 011 11 110 0 011 11 111 0 111 11 101 101 11 100 s= 1 0 0 Syndrome of r (1) (X) r=0 0 1 0 1 1 0 r(X)= X 2 +X 4 +X 5

8. r= 0 0 1 0 1 1 0 101 1 1 101 1 0 00 s= 1 0 1 r (1) = 0 0 0 1 0 1 1 Syndrome Circuit s= 1 0 0 1 extra cycle in the syndrome circuit with input =0 Theorem 1

9. 000 1 1 000 00 100 1 100 00 110 0 011 00 011 1 011 11 110 0 011 11 111 0 111 11 101 101 11 000 s= 0 0 0 r=0 0 1 0 1 1 0 r(X)= X 2 +X 4 +X 5

10. r= 0 0 1 0 1 1 0 101 1 1 101 0 1 00 s= 1 0 1 Syndrome Circuit s= 0 0 0 1 extra cycle in the syndrome circuit with input =1 Theorem 2

11. 0 r= 1 0 1 1 0 1 1 0 1 000 1 100 1 1 11 1 110 011 0 11 11 101 1 1 1 011 101 1 1 1 011 0 101 1 10 0 1 11 101 1 101 1 001 0 0 0 1 1 101 1 101 1 10 0 0 0 10 11101 011 0 0 0 1011101 1 11 0 0 0 101 1 101 1010 1 1 1 101 1 101 0 0 000 0 0 0 101 1 010 000 0 0 0 101 1 010 00 Meggit Decoder

12. DOUBLE ERRORS Store the syndrome of the double error patterns that have an error in the last bit. Cyclically shift the erroneous codeword till one of the two errors becomes in the last bit The syndrome of the shifted codeword matches one of the stored ones Correct the last bit of the codeword Cyclically shift the corrected codeword till the remaining error becomes in the last bit Correct the last bit of the codeword Cyclically shift the corrected codeword till the corrected bits return back to their original location Ex: For a (7,4) Cyclic code, the double error patterns that have 1 in the 7 th bit are 1000001, 0100001, 0010001, 0001001, 0000101, and 0000011