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By Rendong G., Zunquan x., Jinzhi W.

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If r is assigned to the particular decision region Di for which the inner term Pr{r} (1 − Pr{b = bi |r}) is smallest, the word error probability perr will be minimal. Therefore, the decision regions are obtained from the following assignment r ∈ Dj ⇔ Pr{r} 1 − Pr{b = bj |r} = min Pr{r} (1 − Pr{b = bi |r}) . 1≤i≤M Since the probability Pr{r} does not change with index i, this is equivalent to r ∈ Dj ⇔ Pr{b = bj |r} = max Pr{b = bi |r}. 1≤i≤M Finally, we obtain the optimal decoding rule according to ˆ b(r) = bj ⇔ Pr{b = bj |r} = max Pr{b = bi |r}.

0 0 0 0 0 0 0 0 ··· 1 1 1 1 1 1 1 1  0 0 0 0 0 0 0 0 ··· 1 1 1 1 1 1 1 1 is of dimension m × (2m − 1). By suitably rearranging the columns, we obtain the (n − k) × n or m × n parity-check matrix H = Bn−k,k In−k of the equivalent binary Hamming code. The corresponding k × n generator matrix is then given by G = Ik − BTn−k,k . 50 ALGEBRAIC CODING THEORY Because of the number of columns within the parity-check matrix H, the code word length of the binary Hamming code is given by n = 2m − 1 = 2n−k − 1 with the number of binary information symbols k = n − m = 2m − m − 1.

Rn−1 ) can therefore be decoded into the code word bˆ = (r0 , . . , rj −1 , rj + 1, rj +1 , . . , rn−1 ) by calculating bˆj = rj + 1. The weight distribution W (x) of the binary Hamming code H(m) is given by 1 (1 + x)n + n (1 − x)(n+1)/2 (1 + x)(n−1)/2 . n+1 The coefficients wi can be recursively calculated according to W (x) = i wi = n − wi−1 − (n − i + 2) wi−2 i−1 with w0 = 1 and w1 = 0. 29 summarises the properties of the binary Hamming code H(m). 13 In the given matrix the first three columns sum up to the zero column vector of length m.

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