# Binary codes with large distance

Suppose $C \subset \lbrace 0,1\rbrace^{n}$ is a binary code with distance $\delta * n$, for $1/2 < \delta < 1$. Can $|C|$ be arbitrarily large (if I allow n to be arbitrarily large)? Note that the Hadamard code gives you $\delta = 1/2$.

No. If we take $\{-1/\sqrt n,1/\sqrt n\}^n$ instead of $\{0,1\}^n$, the problem reduces to asking if we can have many unit vectors $v_j$ with pairwise scalar products $-\gamma$ or less where $\gamma>0$ is a fixed number. But if we have $N$ such vectors, then the square of the norm of their sum is at most $N-\gamma N(N-1)$. Since this square must be non-negative, we get $N-1\le\gamma^{-1}$ regardless of the dimension.
• When $\delta=1/2,$ the $\gamma$ in the answer is zero. Commented May 20, 2015 at 7:03
• Moreover, if $\delta<1/2$ then the OP's stated requirement is not satisfied. As for Hadamard codes, their size is $M=\log_2(n)$ which is not large at all, but logarithmic in the length. Commented May 20, 2015 at 7:17