I have the $(n,m,d)-$ code $(6,4,4)$ which can clearly be constructed
$$\begin{pmatrix} 000000 \\111100 \\ 001111 \\ 110011 \end{pmatrix}$$
However, if i try using the hamming bound on $m$ i have,
$$m \leq \frac{2^6}{1+\sum^t_{i=1} {n \choose i} }$$ I get the upper bound for $t$ to be $2$, as $$d \geq 2t +1$$ this tells me $m$ must be less than or equal to $2.9$ could someone explain what i'm doing wrong?
If $d \geq 2t+1$ then \begin{align*} t & \leq \lfloor (d-1)/2 \rfloor & \text{since $t$ is a positive integer} \\ &= \lfloor 3/2 \rfloor & \text{since $d=4$} \\ & =1. \end{align*}
So the Hamming Bound is $$m \leq \frac{2^6}{1+\binom{6}{1}} \simeq 9.14$$ which is consistent with $m=4$.