Carryon items are limited to one bag plus one smaller, personal-type item. Southwest Airlines limits carryon bag dimensions to 10x16x24 inches. All Customers and Employees and their items are subject to a thorough physical search.
Your bag would not fit into a 10 x 16 x 24 sizer, so it technically cannot be carried on ( I doubt that any airline will allow it )
To this point, WN gate agents have been pretty lax about enforcing this rule since, with no bag charges, WN passengers tend to have fewer carry-ons than other airlines. However, now that WN planes are moving from 137 seats to 143 seats, I suspect that they will get more strict.