Checkable Codes from Group Rings

We study codes with a single check element derived from group rings, namely, checkable codes. The notion of a code-checkable group ring is introduced. Necessary and sufficient conditions for a group ring to be code-checkable are given in the case where the group is a finite abelian group and the ring is a finite field. This characterization leads to many good examples, among which two checkable codes and two shortened codes have minimum distance better than the lower bound given in Grassl's online table. Furthermore, when a group ring is code-checkable, it is shown that every code in such a group ring admits a generator, and that its dual is also generated by an element which may be deduced directly from a check element of the original code. These are analogous to the generator and parity-check polynomials of cyclic codes. In addition, the structures of reversible and complementary dual checkable codes are established as generalizations of reversible and complementary dual cyclic codes.