I want to cover my garden with ground cover that would stop weeds. I would like something that perennial and not too tall. I prefer something less than 10cm tall but I would consider other options because nothing worse than weeds. Thank you.
Weeds are indeed a nuisance. That being said, weeds do keep bare soil covered and reduce erosion. Choosing ground cover that spreads quickly and grows well in your garden should take over as soil protector. Ground cover with a dense growth pattern should suppress weeds the best.
The “Grow Me Instead” booklet, with link below, is specific to Southern Ontario. It discusses invasive ground covers, English Ivy, Goutweed and Periwinkle, which should be avoided. Alternatives such as Wild Strawberry, Wild Ginger, Wintergreen, Mayapple, Running Euonymus, Foamflower, Wild Geranium Bunchberry and Bearberry are much better choices, although some of these may be too tall for your requirements. There are also photographs beside each writeup which show any flowers that the ground cover may produce.
On page 7 of the pamphlet you will see that Wild Ginger works in a shady location, growing 15-20 cm tall.
Wintergreen is suitable for a woodland habitat garden growing 10- 15 cm tall.
Mayapple would work for a shade; woodland garden but would appear to be much too tall for what you are looking for growing up to 40 cm.
Bunchberry is suitable for a woodland garden growing 7-20 cm tall.
Running Euonymus which grows well in shade, is a low, trailing deciduous shrub which grows up to 45 cm tall. This again would appear taller that you would like.
Foamflower is a good choice for shady woodland or a rock garden growing 13-25 cm tall.
I would not recommend Wild Geranium, which although suited to a shady woodland garden, grows 30-60 cm tall.
Bearberry is suggested for rooftops as well as butterfly and bird gardens. It grows 5-15 cm tall.
Good luck in choosing your ground cover.