The covering radius of extremal self-dual code D11 and its application

D11, constructed by Conway and Sloane, is an extremal self-dual [56,28,12]-code with weight enumerator 1+8190y¹²+622314y¹⁶+11699688y²⁰ +… . We find that the covering radius of D11 is 10. This shows that the inequality R(C₅₆)⩽10 cannot be improved any more, where R(C₅₆) is the covering radius of any extremal doubly even [56,28,12]-code C₅₆ and we use it to get ten new [58,29,10]-codes