The independence number $\alpha(H_q(n,d))$ of the Hamming graph $H_q(n,d)$ is determined by the maximum number $N_q(n,s)$ of $q$-ary sequences of length $n$ that intersect pairwise, i.e., have the same entries, in at least $s=n-d+1$ positions.

• S. El Rouayheb and C.N. Georghiades, Graph Theoretic Methods in Coding Theory (2012)
• R. Ahlswede and L.H. Khachatrian The Diametric Theorem in Hamming Spaces -- Optimal Anticodes (1998)

You want the binary case $q=2$. Closed-form expressions for the number $N_q(n,s)$ exist, see cited references and Theorem 2 of

• P. Frankl and N. Tokushige, The Erdos-Ko–Rado Theorem for Integer Sequences (1999)

The Hamming graph $H(n,d)$ has $2^n$ vertices labeled by the binary vectors of length $n$, two vertices being joined by an edge if and only if the Hamming distance between the corresponding vectors is at least $d$. More generally $H_q(n,d)$ refers to $q^n$ vertices labeled by $q$-ary vectors of length $n$.

• N.J.A. Sloane, Unsolved Problems in Graph Theory Arising from the Study of Codes (1989)

The independence number $\alpha(H_q(n,d))$ of the Hamming graph $H_q(n,d)$ is determined by the maximum number $N_q(n,s)$ of $q$-ary sequences of length $n$ that intersect pairwise, i.e., have the same entries, in at least $s=n-d+1$ positions.

• S. El Rouayheb and C.N. Georghiades, Graph Theoretic Methods in Coding Theory (2012)
• R. Ahlswede and L.H. Khachatrian The Diametric Theorem in Hamming Spaces -- Optimal Anticodes (1998)

You want the binary case $q=2$. Closed-form expressions for the number $N_q(n,s)$ exist, see cited references and Theorem 2 of

• P. Frankl and N. Tokushige, The Erdos-Ko–Rado Theorem for Integer Sequences (1999)

The independence number $\alpha(H_q(n,d))$ of the Hamming graph $H_q(n,d)$ is determined by the maximum number $N_q(n,s)$ of $q$-ary sequences of length $n$ that intersect pairwise, i.e., have the same entries, in at least $s=n-d+1$ positions.

• S. El Rouayheb and C.N. Georghiades, Graph Theoretic Methods in Coding Theory (2012)
• R. Ahlswede and L.H. Khachatrian The Diametric Theorem in Hamming Spaces -- Optimal Anticodes (1998)

You want the binary case $q=2$ and for this case closed-form expressions for this number $N_2(n,s)$ exist (see cited references).

The independence number $\alpha(H_q(n,d))$ of the Hamming graph $H_q(n,d)$ is determined by the maximum number $N_q(n,s)$ of $q$-ary sequences of length $n$ that intersect pairwise, i.e., have the same entries, in at least $s=n-d+1$ positions.

• S. El Rouayheb and C.N. Georghiades, Graph Theoretic Methods in Coding Theory (2012)
• R. Ahlswede and L.H. Khachatrian The Diametric Theorem in Hamming Spaces -- Optimal Anticodes (1998)

You want the binary case $q=2$. Closed-form expressions for the number $N_q(n,s)$ exist, see cited references and Theorem 2 of

• P. Frankl and N. Tokushige, The Erdos-Ko–Rado Theorem for Integer Sequences (1999)