Notes

Encoding

Operation:

• Matrix Multiplication

• How to respresent message data

• How to obtain the generator matrix

MATRIX H (PARITY CHECK MATRIX) - m x n where m is the number of constraints and n is length of the codewords HcT = 0 where cT is the transpose of the codeword

MATRIX G (GENERATOR MATRIX) - k x n where k is the length of the message bits and n is the length of the codeword c = uG where u is the message bits

• How to obtain G from H? H = [A, I_{n-k}] G = [I_k, AT]

Decoding

Operation:

• Sum-product decoding

RECEIVED VECTOR: y CROSSOVER PROBABILITY: p A PRIORI PROBABILITIES VECTOR: r

r_i = log (p/(1-p)) if y_i = 1, = log ((1-p)/p) if y_1 = 0

MESSAGES FROM BIT NODE TO CHECK NODE MATRIX: M_{j,i} where j number of check nodes and i is the number of i nodes

M_{j,i} = sum_{j' belongs to A and j' =/= j}(E_{j',i} + r_i)

Initialization: M_{j,i} = r_i (ATTENTION CHECK THIS INITIALIZATION I BELIEVE THERE IS A CONDITION MISSING WHERE, FOR EACH COLUMN, ONLY THE ROWS THAT HAVE ONE (WHERE THAT BIT NODE IS USED) THE r is substituted and not the whole column as it is presented)

EXTRINSIC PROBABILITIES MATRIX: E

E_{j,i} = (page 32 eq 2.7)

L is the total LLR for each iteration and is given by

L_i = r_i + sum(E_{i,j}) for every checknode j that contains i

z is the current codeword after each iteration where z_i = 1 if L_i < 0 z_i = 0 if L_i > 0

• How to test if the obtained value is or isn't a code word

s = zH' and s must be a vector with only 0's

H' is HT

Example for encoding

[1 0 0 1 0 1] [0 1 0 1 1 1] [0 0 1 0 1 1]

[0 0 0] = [0 0 0 0 0 0] [0 0 1] = [0 0 1 0 1 1] [0 1 0] = [0 1 0 1 1 1] [0 1 1] = [0 1 1 1 0 0] [1 0 0] = [1 0 0 1 0 1] [1 0 1] = [1 0 1 1 1 0] [1 1 0] = [1 1 0 0 1 0] [1 1 1] = [1 1 1 0 0 1]

Ativa as colunas em que a mensagem está a 1 e dá-lhes xor.

Implementação do Mackday por exemplo para o [0 1 1]: pega na segunda linha e copia-a para a codeword [0 1 0 1 1 1] depois pega na terceira linha e da xor das duas [0 1 0 1 1 1] xor [0 0 1 0 1 1] = [0 1 1 1 0 0]