What is the dimensionality of the signal space

Question

Solve the following problem:

Three equiprobable messages m₁, m₂, and m₃ are to be transmitted over an AWGN channel with noise power spectral density ½ N₀. The messages are

 s₁(t) = {1    0≤t≤T                    s₂(t)= -s₃(t) =  { 1        0≤t≤ 1/2T
            {0    otherwise                                       {-1        1/2 T< t ≤ T
                                                                         { 0        otherwise

1. What is the dimensionality of the signal space?

2. Find an appropriate basis for the signal space.

3. Draw the signal constellation for this problem.

4. Derive and sketch the optimal decision regions R₁, R₂, and R₃.

5. Which of the three messages is most vulnerable to errors and why? In other words, which of P(error |mi transmitted), i = 1, 2, 3, is largest?