Quasi-self-reciprocal polynomials and potentially large minimum distance BCH codes

We consider the problem of determining which irreducible polynomials with coefficients in a finite field
GF(q)
are quasi-self-reciprocal. Our characterization of these polynomials is in terms of a set of positive integers,
Dq
.
Dq
has applications to certain BCH codes. This work determines those codes (as specified by
n
, the length of the code, and not by
k
) whose minimum-distance lower bound doubles when the root one is added to the code generator.