  Minimal codewords have applications in decoding linear codes and in
cryptography. We study the maximum number of minimal codewords in binary linear
codes of a given length and dimension. Improved lower and upper bounds on the
maximum number are presented. We determine the exact values for the case of
linear codes of dimension $k$ and length $k+2$ and for small values of the
length and dimension. We also give a formula for the number of minimal
codewords of linear codes of dimension $k$ and length $k+3$.
