Or try a lead link, basically a paternoster set-up where the lead is going to sink into the silt, but the hooklink with a light balanced bait is going to sit up.
You can fish this with your choice of hooklink, mono, fluoro, braid, coated or uncoated, whichever you prefer.
I would recommend a bottom bait that almost floats, I.e drilled out and foamed up in the hole, or even a wafter that only just sinks.
The same lead set up works very well in weed
You may want to play around with the hooklink and lead link length to get it right for your water.
Also, it may be that the carp feed in the silt, not on it, you may need to find a way to get the hookbait at the level they feed at...
Oh, and a quick edit, you may find a simple run ring works as well, and you wouldn't need to use the stop knot shown as per Del Ritchies original pic.