By casing I assume you mean cover and are stripping back a section of cover to leave the skinny core exposed (either to reduce friction on the retrieval end or windage on the hoist end.)

To stop the remaining cover from moving around you can bury it into the core in what is called a core to cover splice. You can find instructions on how to do this on line. Some advice:

• Pick a line which lends itself to this type of work - i.e. the core must be OK to strip (not parallel fibers) and the cover must be spliceable.
• Don't expect to be able to easily do this to an existing halyard which has been loaded up - splicing with the right type of line is easy on new line and virtually impossible on old line - the core and cover become fuzzy - giving too much friction to be able to bury the cover into the core.
• Practice your trade on some scraps of line (using line that has never been used).
• Don't spend good money on a splicing kit from a marine store - they are typically full of fids that are far too big for the type of work you will do on a beach cat. A splicing wand (Brian Toss makes a good one) can be handy, but my crew works his magic with the guts of ball point pens, drill bits, bits of coat hanger - anything thin, smooth and small.
• Beware trying to trim back the cover and seize the cut end to the core. It will look like poop, fray and then jam up on you. Learn to splice.
• An alternative to buying a double braid line and then stripping off that cover you just paid good money for: Use a 12 strand single braid like Swiftcord for the fatter section of line you have to pull on. Splice on skinny 4mm high tech single braid (e.g. Vectran or Spectra) into the end for the sections where you want to save weight, drag and friction. Bury the big line into the little one with a tapered splice and the little one into the big one with a tapered splice.

Here is a link with pictures showing a high tech double braid line being tapered (cover is taper spliced into the core). I would not whip the transition like they show in the pictures - that makes a stiff spot in the line that won't run nicely over a block. Do a little lock stitching to secure a splice - you don't need too much.

Good luck.

Chris.