Classification of Certain Cyclic LCD Codes

Abstract

We show that a necessary and sufficient condition for a cyclic code C of length N over a finite chain ring R (whose maximal ideal has nilpotence 2) to be an LCD code is that C = (f(x)), where f(X) is a self-reciprocal monic divisor of X^N − 1 in R[X] and x = X + (X^N − 1) in R[X]/(X^N − 1). A similar, but slightly different, theorem was proved in 2019 by Z. Liu and J. Wang for general finite chain rings (Theorem 25 in [5]). We provide two proofs, both completely different than the proof of Liu and Wang.

Keywords

• complementary dual (lcd) code
• cyclic codes
• codes over rings