Note that this thread is specifically for an Aronson stack, it may be possible with some other stacks but the red card stuff would definitely be different and the two tiered debruin approach may be difficult or not work with other stacks.

Spoiler alert I have tried the double deBruijn hybrid and it does work on an Aronson deck meaning there are no final duplicates! How easy to use is another question.

Below is my thought process from a few days ago:

I wonder: could a mixture of using just two different back designs coupled with "eyes closed if red" could yield enough information on a four chosen card pattern to pinpoint the first chosen card? This could be a little complex but I think it's doable.

Why do this? well in theory binary is easier than trinary (ternary) but combining two binary searches may or may not be more difficult than one trinary search (cheat sheet).

So let's just look at an Aronson natural binary deBuijn based on red cards being binary "1"s.

0001001010011
1100100110111
1011110101011
0010001001001

Based on four bits there are lots of duplicates in the above wouldbe deBruijn. So how do I combine my ternary debruin with the above debruin? oops, I can't use my ternary because I'm investigating using only two back designs (binary) therefore how do I do this?

Hmmm, maybe use the above binary deBruijn when asking the red card question and have James Grimes 16 bit binary debruin repeated 3.25 times physically on the deck then looking at a two-tiered cheat sheet for the red cards (will have lots of different duplicates but no more than seven which means James Grime's physical binary debruin on the card backs may just work but since it occurs three times may hit one or two instances of combined duplicates?

To do it this way I would have to have a special cheat sheet that I first look for the eyes closed pattern and then refer to that cheat sheet which would then show the appropriate sub patterns based on the backs of the cards (the advantage of this proposed method is that there are just two types of backs instead of three types, using just two types I could use mandolins for zeros and use rider back cards for ones, very easy to read the difference from a long distance, but the disadvantage of this method is you have to phish for the red cards.

Another negative is that the cheat sheet has more stuff, the magician would first have to obtain the red card pattern and find it on the cheat sheet and within that group look for the mandolin back pattern.

How was this combo deBruijn created?
Use an electronic spreadsheet such as the free Google Sheets to do the following steps:

First highlight columns A thru F and format them as text (this well insure that your entries will be left justified and not have leading zeros stripped off).

Under column C list the cards in your stack (in cell C1 thru C52).
Tamariz mnemonica example:
4C
2H
7D
3C
4H
.
.
. (below are cells C49 thru C52)
TD
6C
aH
9D

Leave column B blank for now.

Under column A of your spreadsheet list the 52 patterns based on your stack (the red cards in your stack will be binary "1" and the black cards will be binary "0"). The easy way to do this is to first list the single digit (1 or 0) in rows 1 thru 52.
Using Tamariz (mnemonica) as an example then you put "0" in cell A1 (because the four of clubs is black), you put "1" in cell A2 (because the two of hearts is red), and continue doing this right down the A column for all 52 rows.

So now your column A (Tamariz example):
0
1
1
0
1
.
.
. (below are cells A49 thru A52)
1
0
1
1
Next you will update those same cells (A1 thru A52) by adding three digits (composed of 0 or 1). You do this by looking at the three cells in column A immediately below the cell you are updating and appending those three digits to the one digit that is already in the cell you are updating.

Tamariz example of cell A1:
you already have a "0" in cell A1 then you append the values that are in cells A2, A3, A4 so that you now have 0110 in cell A1.

Tamariz example of cell A2:
you already have a "1" in cell A2 then you append the values that are in cells A3, A4, A5 so that you now have 1101 in cell A2.

You do this process until all 52 cells in column A have 4 digits. Note that when you get near the bottom ie cells A50, A51, A52 you wrap around to cells A1, A2, A3 copying the first digit only to get your extra three digits.

Column A (Tamariz example) will now look like this:

0110
1101
1011
0110
1101
.
.
. (below are cells A49 thru A52)
1011
0110
1101
1011

You are finished creating column A. Column A will not be changed.

Don't worry about duplicate patterns in column A, this is okay, you will have quite a few.

Now for column B:
Use the James Grime's 16 bit debruin 4 bit patterns (use it regardless of which memorized stack you are using).
1111 0000 1001 1010

And since these patterns are fixed I'll just list them here:
Initializing column B (cells B1 thru B16)
1
1
1
1
0
0
0
0
1
0
0
1
1
0
1
0

Next expand the values in cells B1 thru B16 to 4 digit patterns (by appending the values in the next three cells below it):
1111
1110
1100
1000
0000
0001
0010
0100
1001
0011
0110
1101
1010
0101
1011
0111
Note that cells B14, B15, B16 grab their extra digit(s) from the first digit in the cells B1, B2, B3).

Next copy and paste the above B1 thru B16 cells into B17 through B32.

Now copy and paste the above B1 thru B16 cells into cells B33 through B48.

Finally in cells B49 through B52 put the following 4 digit patterns:
0000
0001
0011
0111
Cell B49 has 0000, cell B50 has 0001, cell B51 has 0011, cell B52 has 0111.

Now columns A, B, C are complete so copy them over to columns D, E, F so we can sort them (we want to keep columns A, B, C pristine).

How to sort:
Highlight columns D, E, F
Choose the sort option (in Google Sheets it's under data, sort, advanced).
For the sort fields select column D then select column E (column F has the two character card names and is not selected as a sort field but since it was highlighted it's coming along for the ride as it must). Actually it will not hurt anything if column F is selected as a third sort field but it won't really do anything because columns D and E dominate the sort.

Click the do it button, okay button, sort button or whatever button your spreadsheet wants you to click.

Now columns D, E, F are in the proper order for your cheat sheet so highlight them and copy and paste them to Google docs or someplace you can manipulate the rows a little bit.

You need to add a blank row when the data in column D changes (you will have a lot of duplicates in column D so add a blank row immediately before the row that is different in column D). This will make your cheat sheet much much easier to read.

So print off your cheat sheet and put it probably in a funny looking booklet that says something like "mentalism for dummies" or "Einstein's Instructions for Magicians" or "Wikopedia must be Right" or "Farmers Almanac Unlimited". Note you can print your own cover sheet and glue it to a lightweight booklet.

Anyhow back to the cheat sheet itself. When performing the trick you say "raise your hand if you have a red card" and based on the pattern
(a raised hand represents the digit 1 whereas no hand represents the digit 0) so you look for that pattern in column D and then you perhaps put a finger on that first row.

I digress to explain the backs of the cards. Have your mem deck in order using all bicycle rider back cards. Then look in column B of your spreadsheet and everywhere that the first digit is a "0" replace the corresponding card with a mandolin back card (of course of the same suit and value and same color back).
Now back to the performance:

Then you look at the 4 cards themselves (have the 4 spectators hold their card faces to their forehead and concentrate). You the magician identify a four digit pattern amongst the four cards where the mandolin back cards represent the digit 0 and the rider back cards represent the digit 1.

Using the 4 card pattern, you then look at your cheat sheet column E (within the D group that you have your finger on) to find that pattern and then immediately to the right in column F is the two character identity of the first card of the four.

Final thoughts:
Once you have the deck constructed and the cheat sheet printed off the performance rules (even though two tiered) are pretty straightforward:
1. determine the red card pattern from the four spectators and find that pattern in column D
2. determine the physical four card pattern (Rider equals 1, Mandolin equals 0) and find that pattern in column E (within the D group). The identity of the first card of the four is to the right (in column F). Wah Lah!

Compared to other methods that ask for "who has red cards" this 4 card hybrid combo method has some advantages and disadvantages.

The deBruijn methods that have six cards (six spectators) reveal whether they have red cards has the disadvantage that all it takes is just one spectator to mess up and it can ruin your trick whereas with this hybrid 4 card method there are less spectators to mess things up. Also my hybrid method can easily be done with two spectators (one card for each hand).

There are 4 card deBruijn methods that use 16 different cards that are repeated two more times. The disadvantage of that method is the face up deck cannot be scrutinized (most people will not want to and even if they did they probably would not detect the duplications however it is a small negative for that type of deck whereas my hybrid method removes that negative because the faces of the cards can be thoroughly examined because it has 52 unique normal playing card faces).

The biggest con of my hybrid combo method is that the cheat sheet is more involved. Another con is that a spectator may notice the backs of the cards being different but that can be covered by saying "kids did it, I'll fix it tomorrow". Another solution to this con is to use all bicycle Rider back cards and dull the cherubs in the four corners on the cards that represent 0 (I talk about this "dulling" technique towards the end of a separate thread titled "Need ternary 4 bit deBruijn").

The biggest con to my hybrid combo method is that if I'm going to use some markings on cards then why not just use my ternary deBruijn (see another thread) and not have to use a phishing statement for the red cards.

This hybrid combo method has one slight pro over my ternary deBruijn and that is it uses less marked cards (28 vs about 34) also this hybrid combo method uses only two types of backs whereas the ternary deBruijn uses three types of backs.

All in all I like my ternary deBruijn method (see another thread) much better than this complex hybrid combo method because the ternary deBruijn is much simpler to execute and less things to go wrong (no red card phish).

Well this thread may not be a total loss because the hybrid combo method might have application somewhere else and this could tickle the gray cells for somebody else trying to solve a particular problem.