What is the optimum maximum-likelihood detector

Question

Response to the following problem:

The discrete sequence

r_k = √ε_ck + n_k,          k = 1,2,....,n

represents the output sequence of samples from a demodulator, where c_k = ±1 are elements of one of two possible code words, c₁ = [1 1 ··· 1] and c₂ = [1 1 ··· 1 -1 ··· -1]. The code word c₂ has w elements that are +1 and n - w elements that are -1, where w is some positive integer. The noise sequence {nk} is white Gaussian with variance σ².

1. What is the optimum maximum-likelihood detector for the two possible transmitted signals?

2. Determine the probability of error as a function of the parameters (σ², E, w).

3. What is the value of w that minimizes the error?