How to generate Moving Average model

In order to generate Autoregressive model, we have the aryule() command and we can also use filtersEstimating AR model. But how do I generate MA model? For instance, can somebody please show how to generate MA(20) model? I could not find any appropriate technique to do so. The noise is generated from a nonlinear map

epsilon(1) = 0.01;

for i =1 : N 
        epsilon(i+1) = 4*epsilon(i)*(1-epsilon(i));
    end

So, the MA model will regress over epsilon terms.

Q1: Shall be extremely helpful if the code and functional form of an MA model is shown preferably MA(20) using the above noise model.

Q2: This is how I generated an AR(20) using random noise but don't know how to use the above equation as the noise instead of using rand for both MA and AR

 %Generate sine wave = A*sin(2*pi*f*t + phi)
t = linspace(0,1,1000);
A = 5;
f = 2;
phi = pi/8;
sinewave = A*sin(2*pi*f*t + phi);
noisy_sine=sinewave+0.5*randn(size(t));
subplot(1,2,1);
plot(t, sinewave)
hold on;
subplot(1,2,2);
plot(t,noisy_sine);

%Generate AR model(20)
order =20;
ARCoeff = aryule(noisy_sine,order);

I had a same problem. The following links can help :

https://www.mathworks.com/examples/econometrics/mw/econ-ex18477389-simulate-an-ma-process

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Simulate an MA Process

This example shows how to simulate sample paths from a stationary MA(12) process without specifying presample observations.

Contents

1. Specify a model.
2. Generate sample paths.
3. Plot the simulation variance.

Step 1. Specify a model.

Specify the MA(12) model

where the innovation distribution is Gaussian with variance 0.2.

model = arima('Constant',0.5,'MA',{0.8,0.2},...
              'MALags',[1,12],'Variance',0.2);

Step 2. Generate sample paths. Generate 200 sample paths, each with 60 observations.

rng('default')
Y = simulate(model,60,'NumPaths',200);

Step 3. Plot the simulation variance.

figure
plot(Y,'Color',[.85,.85,.85])
hold on
h = plot(mean(Y,2),'k','LineWidth',2)
legend(h,'Simulation Mean','Location','NorthWest')
title('MA(12) Process')
hold off