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White Led

How to make superbright White LED as bright as possible?

I'm using power supply of 240V AC, going thru a step down transformer to 12V AC, thru a brigde rectifier and across a capacitor. I connect 4 LEDs in series together with a 1K ohm resistor. How to make these LEDs as bright as possible?

If you have a data sheet for your LEDs it would be helpful, while the 20ma value others have given is a reasonable value, if you are looking for maximum brightness you may want to know what your specific LED are rated for, if they are "super bright" the current rating might be higher. You will also need to know the forward voltage at the current you want to use.
Perhaps a typical forward voltage for a white LED might be 3.4V, with 4 in series you have 13.6V, which is more than 12, that might be why some said parallel would be better, but actually you probably have more than 12V, a very lightly loaded bridge rectifier circuit will be charging the filter capacitor to near the peak voltage, not the RMS, 12 * sqrt(2) ≈ 17V subtracting 1.4V for the drop in the bridge, and rounding down since you won't quite get to the peak, lets call the available voltage about 15V
15 - 13.6 = 1.4V; 1.4V / 20ma = 70Ω
Now 1.4V is a pretty small difference, and if your particular LEDs have a little higher voltage, or your transformer or capacitor is small you might not get maximum brightness. So you might put two series pairs in parallel, but each series pair will need it's own series resistor, putting a number of paralleled LEDs in series with one resistor does not work well, since there is some slight variation in LEDs the current will not be equal in all of them (while if they are in series the current has to be same, by definition)
But if you can use a series circuit it will be the most efficient, in the example above, 1.4V * 20mA = 28mW lost, if you used all 4 LEDs in parallel, with a 15 - 3.4 = 11.6V drop needed (580Ω) 11.6 * 4 * 20mA = 928mW loss, not a lot of powers, but still..

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