I expect not. If I get your question, you are asking for a basis for R, the row space of G, that
has as many zero coordinates as possible. I can imagine that there are subspaces of dimension m inside
F^2m for m small with respect to the cardinality of the base field F that avoid bases with more than the minimal
number, but I cannot
at this time provide explicit examples with proof for all n. Something that should work is having the ith row of
G be e_i concatenated with f_i, where e_i is the canonical ith basis vector with m-1 zeros and one 1,
and f_i is an m vector of values of (x^(i-1) +i) evaluated at x=1 to m, but I have only looked at the case
m=2, where the rows are 1 0 2 2 and 0 1 3 4. The argument should go something like : if I produce
k zeroes in "the second half" of such a G, I take away k or more zeros in "the first half" because the
coefficients of the polynomial which is generated in the second half will dictate many nonzero entries
in the first half, but I have not done the whole argument myself.

Gerhard "Ask Me About System Design" Paseman, 2013.01.15