I think this does the job, but it needs testing on some more demanding sample data.

c:\test>600418.pl
This input

        0 0 1 1 0
        1 0 1 1 1
        0 1 0 1 0
        1 1 0 0 1
        0 1 1 1 0

Inverted looks like this

        0 : 01010
        1 : 00111
        2 : 11001
        3 : 11101
        4 : 01010

This subset can be removed from the main group:

        0 : 01010
        4 : 01010

c:\test>600418.pl
This input

        0 0 0 1 1 0 0 0
        0 1 0 1 1 1 1 0
        1 0 1 0 1 0 1 1
        1 1 1 0 0 1 1 1
        1 0 1 1 1 0 0 1

Inverted looks like this

        0 : 00111
        1 : 01010
        2 : 00111
        3 : 11001
        4 : 11101
        5 : 01010
        6 : 01110
        7 : 00111

This subset can be removed from the main group:

        1 : 01010
        5 : 01010

This subset can be removed from the main group:

        0 : 00111
        2 : 00111
        7 : 00111
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Relating the subset numbering back to the pre-inversion set is left as an exercise.

#! perl -slw
use strict;

=comment
my @grid = (
    [ 0,   0,   1,   1,   0 ],
    [ 1,   0,   1,   1,   1 ],
    [ 0,   1,   0,   1,   0 ],
    [ 1,   1,   0,   0,   1 ],
    [ 0,   1,   1,   1,   0 ],
);
=cut

my @grid = (
    # 0  1  2  3  4  5  6  7
    [ 0, 0, 0, 1, 1, 0, 0, 0, ],
    [ 0, 1, 0, 1, 1, 1, 1, 0, ],
    [ 1, 0, 1, 0, 1, 0, 1, 1, ],
    [ 1, 1, 1, 0, 0, 1, 1, 1, ],
    [ 1, 0, 1, 1, 1, 0, 0, 1, ],
);
print "This input\n";
print "\t@$_" for @grid;

my @inverted;
for my $i ( 0 .. $#grid ) {
    $inverted[ $_ ] .= $grid[ $i ][ $_ ] for 0 .. $#{ $grid[ $i ] };
}
print "\nInverted looks like this\n";
print "\t$_ : $inverted[ $_ ]" for 0 .. $#inverted;

my @bitCounts;
push @{ $bitCounts[ $inverted[ $_ ] =~ tr[1][1] ] }, $_ for 0 .. $#inv
+erted;

#print @{ $_||[] } for @bitCounts;

for my $c ( 2 .. $#bitCounts ) {
    #print( "skipping $c elements < ${ \ scalar @{ $bitCounts[ $c ] } 
+} bits" ),
    next unless @{ $bitCounts[ $c ] } >= $c;

    my @set = @{ $bitCounts[ $c ] };

    for my $offset ( 0 .. @set - $c ) {

        my $matches = grep{
            $inverted[ $set[ $offset ] ] eq $_
        } @inverted[ @set[ $offset .. $#set ] ];
        #print "set:@set matches:$matches";

        last unless $c <= $matches;

        print "\nThis subset can be removed from the main group:\n";
        print "\t$_ : $inverted[ $_ ]" for grep{
            $inverted[ $set[ $offset ] ] eq $inverted[ $_ ]
        } @set[ $offset .. $#set ];
    }
}
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