Simple Methods for Deriving Lower Bounds in Coding Theory

We consider various methods of deriving lower bounds on the cardinality of a code. A new asymptotic expression is obtained for the maximum cardinality $L(n,t)$ of a length-$n$ binary code correcting $t$ localized errors:
$$ L(n,t)=\frac{2^n}{\sum^t_{j=0}C_n^j}(1+o(1)),\ \ t^3 n^{-1}\ln n\to 0. $$