  The problem of finding subfield subcodes of generalized Reed-Solomon (GRS)
codes (i.e., alternant codes) is considered. A pure linear algebraic approach
is taken in order to derive message constraints that generalize the well known
conjugacy constraints for cyclic GRS codes and their Bose-Chaudhuri-Hocquenghem
(BCH) subfield subcodes. It is shown that the presented technique can be used
for finding nested subfield subcodes with increasing design distance.
