Hi Shabar,
To be honest I would just use a heavy lead!
And cast around as much of the lake as you can to find some spots" hard or clear" when it's not busy As these have been cleared by feeding fish, find these and they will produce Carp, and these areas will be there waiting to be found and any good rig will work on theses areas, Chod rigs will work "over" silt, so will lighter leads and longer hooklinks( be carefull, not too long if baiting with a particle, as you could be biten off) Another method I use in silt is after casting, put tension from the rod tip until you feel the lead pull free, this will bring the lead onto the silt and not deep in it(light lead) again, give it a go