I am working with a program in which some arrays must be accessed based on their names.

You really don't want to do that. No, I mean you really don't want to do that. use strict makes symbolic references illegal for a reason.

There are a tiny number of advanced techniques where you need to use symbolic references. This isn't one of them.

In this case, what you really want is a hash.

#!/usr/bin/perl

use strict;
use warnings;

my %descriptor_array = (
                        1 => [ 1, 2, 3, 4 ],
                        2 => [ 1, 2, 3, 4 ],
                        3 => [ 1, 2, 3, 4 ],
                        4 => [ 1, 2, 3, 4 ],
                       );

foreach my $array_no (1 .. 4) {
  print "$array_no: @{$descriptor_array{$array_no}}\n";
}
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If you alter your code to use a model like this then you'll be far happier.

perldsc will probably be useful to you.