The most common cause of conifer browning is wind damage, and where they are damaged right back to the woody stem they're unlikely to ever regrow from those areas - this means it could spread further out.
If that part of your hedge catches the winds then i'd say that was the cause.

There ARE diseases which affect conifers and the tree would die right back, it would be an idea to send samples to the RHS to get them to confirm the presence of disease.

It looks to me as though it could be damage caused by the conifer aphid, check for signs of sooty mould aprx three inches in from the tips, also check all other parts of the hedge for this, if you have the conifer aphid, or conifer scale then spray with provado ultimate bugclear,where it has died back it wont grow back, however if you let the surounding areas grow out a little then you can cleverly wire them across into the brown areas and given a little time you can regreen these areas.