  In this article we investigate Berlekamp's negacyclic codes and discover that
these codes, when considered over the integers modulo 4, do not suffer any of
the restrictions on the minimum distance observed in Berlekamp's original
papers. The codes considered here have minimim Lee distance at least 2t+1,
where the generator polynomial of the code has roots z,z^3,...,z^{2t+1} for a
primitive 2nth root of unity z in a Galois extension of Z4. No restriction on t
is imposed. We present an algebraic decoding algorithm for this class of codes
that corrects any error pattern of Lee weight at most t. Our treatment uses
Grobner bases and the decoding complexity is quadratic in t.
