Characterization theorems for extending Goppa codes to cyclic codes (Corresp.)

Several theorems are presented which characterize Goppa codes having the property of becoming cyclic when an overall parity cheek is added. If such a Goppa code has location set
L = GF (q^{m})
and a Goppa polynomial
g(z)
that is irreducible over
GF(q^{m})
, then
g(z)
must be a quadratic. Goppa codes defined by
(z- \beta)^{a}
and location set
L
with cardinality
n
such that
n+l|q^{m}-1
are considered along with their subcodes. A sufficient condition on
L
is derived for the extended codes to become cyclic. This condition is also necessary when
a
= 1. The construction of
L
for different
n
satisfying the stated condition is investigated in some detail. Some irreversible Goppa codes have been shown to become cyclic when extended by an overall parity check.