Dual product constructions of Reed - Muller type codes (Corresp.)

Various linear and nonlinear
R(r,m)
codes having parameters
(2^{m}, 2^{k}, 2^{m-r})
with
k=\sum_{i=0}^{r}\left(^{m}_{i}\right)
are constructed from
R(r,q)
and
R(r,p)
codes,
m=p+q
. A dual construction for
R(m-r,m)
codes from
R(p-r,p)
and
R(q-r,q)
codes is also presented,
m=p+q
. As a simple corollary we have that the number of nonequivalent
R(r,m)
codes is at least exponential in the length (for
r>1)
. For
R(m-r,m)
codes, the lower bound is doubly exponential in the length (for
r>1)
.