Parameters of the Reed-Muller code

Question

I saw the parameters of RM$_{m,t}$ are $[n,k,d]=[2^m,\sum_{i=0}^{t}{m \choose i},2^{m-t}]$ I can't work out why the dimension is $\sum_{i=0}^{t}{m \choose i}$. A nudge in the right direction would be appreciated.

Answer 1

The codewords of a Reed-Muller code correspond to polynomials of degree at most $t$ in $x_1,\ldots,x_m$ with the restriction that each variable appears to at most the first power. The dimension of these under the extra condition that each term has $i$ $x_j$s is $\binom mi$.