Cats are extraordinarily fastidious creatures. Outdoors they tend to urinate and defecate in relatively open and previously unused areas. Unless they are "marking" - deliberately depositing the scent of their urine or faeces as a chemical "autograph" for other cats - they carefully cover their waste and move on.
It is asking quite a lot, then, to provide a single litter tray in a home with several cats, to fill it with litter that is unpleasant to step into, clean it only infrequently, and expect cats to use it several times daily, day after day. It's no surprise that the most common behaviour problem reported by cat owners is urination and defecation outside the litter tray.
To minimize the chances that your cat will christen the carpet, keep in mind the habits of wild cats. A good "rule of paw" is to have one litter tray available for each cat, plus one extra. For example, if you have one cat, you should provide two litter trays; if you have two cats, you should provide three litter trays. Litter trays should be placed in accessible areas - avoid dark, damp cellars and distant bedrooms, or areas unprotected from the noises of washing machines and furnaces.
To accommodate the preference for open spaces (and dislike for odours), avoid covered litter trays (some cats do prefer the privacy that covered litter trays provide). Instead, use as large a litter tray as possible. A study of substrate preferences has shown that cats prefer the clumping, scoopable litters to other types.
Finally, try to keep your litter trays as clean as possible, removing urine and faeces at least daily. These efforts will certainly be rewarded by your obliging cat.