Error Message with Maximize

So, I wanted to do the following maximization, from the article https://arxiv.org/abs/1706.06221
"Non-asymptotic entanglement distillation", which can be seen below:

And they used mathematica for solving the problem, with great success, as they stated "This linear program can be solved exactly via Mathematica.", and were able to make the following plot with d = 3, F = 0.9, and error tolerance 0.001 (epsilon)

However, when tried to do it with the same parameters for d, F and epsilon:

    `d = 3;
     F = 0.9;
     n = 13;
     epsilon = 0.001;
     Maximize[{1/eta, 
     Sum[Binomial[n, i ]*F^i*(1 - F)^(n - i)*m[i], {i, 0, n}] >= 1 - epsilon  
     && (-1)*eta <= Sum[x[i, 0]*m[i], {i, 0, n}] <= eta 
     && (-1)*eta <= Sum[x[i, 1]*m[i], {i, 0, n}] <= eta 
     && (-1)*eta <= Sum[x[i, 2]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 3]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 4]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 5]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 6]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 7]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 8]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 9]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 10]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 11]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 12]*m[i], {i, 0, n}] <= eta
     && (-1)*eta <= Sum[x[i, 13]*m[i], {i, 0, n}] <= eta && 
     0 <= m[0] && m[0] <= 1 && 0 <= m[1] <= 1 && 0 <= m[2] <= 1 && 
     0 <= m[3] <= 1 && 0 <= m[4] <= 1 && 0 <= m[5] <= 1 && 
     0 <= m[6] <= 1 && 0 <= m[7] <= 1 && 0 <= m[8] <= 1 && 
     0 <= m[9] <= 1 && 0 <= m[10] <= 1 && 0 <= m[11] <= 1 && 
     0 <= m[12] <= 1 && 0 <= m[13] <= 1}, {m[0], m[1], m[2], m[3], m[4],
     m[5], m[6], m[7], m[8], m[9], m[10], m[11], m[12], m[13], eta}]

`

Mathematica gave me the following error message, depending on the value of n. For example, I was able to do it for n=20, but no for n=13, the same way I obtained success for n=32, but got the error message for n=24. Changing F and d also affects if I am able to do the calculation or get the error message.

        NMaximize::nsol: There are no points that satisfy the constraints {1.*10^-13 m[0]+1.17*10^-11 
        m[1]+6.318*10^-10 m[2]+2.08494*10^-8 m[3]+4.69111*10^-7 m[4]+7.59961*10^-6 m[5]+0.0000911953 
        m[6]+0.000820757 m[7]+0.00554011 m[8]+0.0277006 m[9]+0.099722 m[10]+0.244772 m[11]+0.367158 
        m[12]+0.254187 m[13]>=0.999,<<49>>,<<7>>}.

and

        Maximize::infeas: There are no values of 
        {m[0],m[1],m[2],m[3],m[4],m[5],m[6],m[7],m[8],m[9],m[10],m[11],m[12],m[13],eta} for which the 
        constraints <<1>> are satisfied and the objective function 1/eta is real-valued. 

I tried using NMaximize, as well as SetPrecision to Machine Precision, but it still gave me the same error message. Does anyone know how to overcome this problem?