ok, so the 2^N terms are each a product of N terms, a[i][j], where j can be 0 or 1 and i goes from 0 to N-1
ie, each one looks like, say
 a[0][0] * a[1][0] * a[2][1] * a[3][0] * ... * a[N-1][1]
So, we want do some binary magic.
how about something like:

#!/usr/bin/perl -wl
use strict;
use Data::Dumper;

my $N = 4;
my $size = 32;
my $arr = [[1, 2],
           [3, 4],
           [5, 6],
           [7, 8]];

my @C;

foreach my $m (0..2**$N - 1) {

    my $vec = "";
    vec($vec, 0, $size) = $m;
    my @bits = split(//, unpack("B*", $vec));
    splice(@bits, 0, $size - $N);

    # so @bits now contains the binary expansion of $m
    # with exactly $N bits, as you can verify by uncommenting
#    print join "::", @bits;


    $C[ $m ] = 1;
    # now we multiply the $N terms $arr->[i][j]
    # where j is the i-th bit in the binary expansion of $m
    $C[ $m ] *= $arr->[ $_ ][ $bits[$_] ] for 0..$N-1;
}

print Dumper \@C;
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note, $size must be a power of 2, for vec to work, and unless you're on a 64-bit machine, it probably cannot be more than 32.

also, this works for $N no more than 31(32?) - but you may have other problems (memory say), if you want to go above that.

Hope this helps.