That's a De Bruijn sequence. See https://en.wikipedia.org/wiki/De_Bruijn_sequence https://en.wikipedia.org/wiki/De_Bruijn_sequence . A example applied to door c

That's a De Bruijn sequence. See https://en.wikipedia.org/wiki/De_Bruijn_sequence https://en.wikipedia.org/wiki/De_Bruijn_sequence . A example applied to door codes is at http://stefangeens.com/2001-2013/2004/10/the-de-bruijn-code/ http://stefangeens.com/2001-2013/2004/10/the-de-bruijn-code/ . A search for "De Bruijn garage door" finds a couple of examples of what you mentioned.