  Double Toeplitz (DT) codes are codes with a generator matrix of the form
$(I,T)$ with $T$ a Toeplitz matrix, that is to say constant on the diagonals
parallel to the main. When $T$ is tridiagonal and symmetric we determine its
spectrum explicitly by using Dickson polynomials, and deduce from there
conditions for the code to be LCD. Using a special concatenation process, we
construct optimal or quasi-optimal examples of binary and ternary LCD codes
from DT codes over extension fields.
