On a class of convolutional codes

For the case when
k
divides
n
, we introduce a special class of
(n,k)
F
-ary convolutional codes,
F=
GF
(q)
a finite field, by considering the input to an
(n,k)
encoder as a sequence over GF
(q^{k})
, the output as a sequence over GF
(q^{n})
(an idea first used by Dym [10]), and then considering encoders which correspond to convolving the input with a fixed sequence
\Gamma_{0}, \Gamma_{1}, \cdots \Gamma_{m}
over GF
(q^{n})
. A means of obtaining an encoder
G(D)
from the polynomial
\Gamma(D)=\Gamma_{0}+\Gamma_{1}D+\cdots +\Gamma_{m}D^{m}
with respect to a basis for GF
(q^{n})
over GF
(q)
is described. A criterion on
\Gamma(D)
in order for any
G(D)
obtained from it to be noncatastrophic is established, which involves computing only the greatest common divisor (gcd) among
s=n/k
polynomials over GF
(q^{k})
. This criterion is shown to coincide with that of Massey and Sain when
k=1
. It is shown that if
\Gamma(D)
is noncatastrophic (i.e., if encoders obtained from it are noncatastrophic) and has zero delay, then any encoder
G(D)
obtained from it is minimal and has a zero-delay feed-forward inverse. The number of zero-delay noncatastrophic polynomials over GF
(q^{n})
of degree
m
is shown to be
q^{nm}(q^{n}-1)(q^{n-k}-1)/q^{n-k}(q^{k}-1)
, a formula which coincides with that of Shusta [11] when
k=1
. The class of codes just described is shown to form a group under multiplication. If the basis is normal, the class is shown to be dosed under cyclic shifting. When
k=1
the class of codes described coincides with the class of all
(n,1)
F
-ary convolutional codes; hence we obtain new proofs of certain well-known results about this latter class of codes. Finally, the binary rate
1/2
convolutional codes obtained from the noncatastrophic divisors of
D^{15}+1
over GF
(2^{2})
are studied and optimal codes of constraint lengths
6, 8
, and
12
found.