MEGGIT DECODER

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 n error patterns of single error

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

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= v= e= r= ES E= S=1 0 1

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

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= r(X)= X 2 +X 4 +X s=1 0 1

r (1) = r (1) (X)= X 3 +X 5 +X s= Syndrome of r (1) (X) r= r(X)= X 2 +X 4 +X 5

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

s= r= r(X)= X 2 +X 4 +X 5

r= s= Syndrome Circuit s= extra cycle in the syndrome circuit with input =1 Theorem 2

0 r= Meggit Decoder

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 , , , , , and