  This paper provides a combinatorial characterization of weak algebraic
manipulation detection (AMD) codes via a kind of generalized external
difference families called bounded standard weighted external difference
families (BSWEDFs). By means of this characterization, we improve a known lower
bound on the maximum probability of successful tampering for the adversary's
all possible strategies in weak AMD codes. We clarify the relationship between
weak AMD codes and BSWEDFs with various properties. We also propose several
explicit constructions for BSWEDFs, some of which can generate new optimal weak
AMD codes.
