Ok...so we'll call your strip mostly shady. Ground covers that will do well in shade:

Sweet Woodruff
Pachysandra
Wild Ginger (Asarum)
Snow on the Mountain (only if you are sure you want the entire area covered with this forever!!!)
Lily of the Valley...same as above...
Houttuynia (Chameleon Plant)...deer won't eat it, but ditto the above 2...
Epimedium...some spread, some clump
Lysimachia nummularia 'Aurea'
Tiarella...some spread
Lamium

There are probably others, but this should give you somewhere to start.

Why do you want ground covers, though? 4' wide is enough room for an actual garden and it would be so much more interesting than a carpet of one or two plant varieties.

You could add daylilies for color. If this is not an area you go to or look at frequently you could use native daylilies (aka ditch lilies) which will eventually shade the grass out. I have some to share - along with lily of the valley and snow-on-the-mountain.

Just a thought.

Cricket