In Field $\mathbb{F}_{256}$ we are given homogeneous equations in 255 variables from the field. It is said that maximum number of linearly independent such equations we can get are 247. Why ?
To be more general,
In Field $\mathbb{F}_{2^m}$ we are given homogeneous equations in 2^m-1 variables from the field. It is said that maximum number of linearly independent such equations we can get are $2^m -m-1$. Why ?