  A de Bruijn covering code is a q-ary string S so that every q-ary string is
at most R symbol changes from some n-word appearing consecutively in S. We
introduce these codes and prove that they can have length close to the smallest
possible covering code. The proof employs tools from field theory, probability,
and linear algebra. We also prove a number of ``spectral'' results on de Bruijn
covering codes. Included is a table of the best known bounds on the lengths of
small binary de Bruijn covering codes, up to R=11 and n=13, followed by several
open questions in this area.
