LCD codes over finite chain rings

• Under suitable conditions, we provide a necessary and sufficient condition for LCD linear codes over finite chain rings.
• We characterize LCD linear codes over finite principal ideal rings.
• We give an example of LCD code over finite principal ideal rings to be a good code.

A linear code with a complementary-dual (an LCD code) is defined to be a linear code C   satisfying C∩C⊥={0}C∩C⊥={0}. We provide a necessary condition for an LCD linear code C over a finite chain ring. Under suitable conditions, we give a sufficient condition under which a linear code C over a finite chain ring is LCD. In particular, we derive a necessary and sufficient condition for free linear codes over a finite chain ring to be LCD. We also give a characterization of LCD codes over principal ideal rings.