  LCD codes are linear codes with important cryptographic applications.
Recently, a method has been presented to transform any linear code into an LCD
code with the same parameters when it is supported on a finite field with
cardinality larger than 3. Hence, the study of LCD codes is mainly open for
binary and ternary fields. Subfield-subcodes of $J$-affine variety codes are a
generalization of BCH codes which have been successfully used for constructing
good quantum codes. We describe binary and ternary LCD codes constructed as
subfield-subcodes of $J$-affine variety codes and provide some new and good LCD
codes coming from this construction.
