I have used cereal rye alone or mixed with wheat as part of a cover crop planting for going on 4 years. I always just cut it with the weed eater about a week before I need to plant out. I then put a thick layer of shredded leaf mulch over it. I have never had a problem with it growing back. Some plants had seed heads, others don't. By September, when I reach under the leaf mulch, I can't find any trace of the previous winter's cover crop. I don't till, strictly no till. Good luck.
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