as for the length of the coil, longer is better.
try for at least 25 of small tubing... 1/4 ID
tubing at my ace is $0.20, so 25 feet of it would only be $5
... it is the semi ridged white stuff, not the clear - should hold up to being coiled better than the soft clear stuff.
most of the designs i've seen use a 4" PVC
pipe to form and contain the coil and like you said, bioball media.
the objective of the long tube is to deplete the oxygen, releasing a stream of nitrate rich oxygen poor water in the main biochamber, where the bioballs play home to the denitrafication bacteria. Because the volume is much greater, the water moves a lot slower in the biochamber, and thats where the bacteria want to grow. Water then exits the biochamber and is remixed with tank water in a sump, or over the rocks like we discussed.
I speculate that the tube and extra biochamber take some of the guess work out of determining flowrate and tube length. a coil-only system would need an incredibly low flow rate of only a few gallons per hour, due to the small volume.
adding the biochamber and biomedia provide an enormous increase in the surface area, allowing more bacteria to grow and process the water faster.
Check out this link for more info