I am interested in whether the following problem is known.

For a given binary vector $V$ of length $n\geq m$, let $S$ be a subset of the possible subvectors of $V$ of length $m$ and say that the size of $S$ is simply the number of subvectors it contains.

For fixed $n$ and $m$ and maximising over all possible binary vectors of length $n$, what is the maximum size $S$ that has the property that the Hamming distance between all pairs $v_1,v_2 \in S$ is at least $d$?

If subvectors were not allowed to overlap then this would be a basic coding theory question.

(Clarification: A subvector has consecutive coordinates.)

**Examples.**

Let us simplify by setting n = 2m-1 so there are $m$ subvectors of a fixed $V$ each with length $m$.

Set $n = 7$, $m = 4$ and $d=2$. The vector $(1, 0, 1, 0, 0, 1, 1)$ has subvectors $(1,0,1,0), (0,1,0,0), (1,0,0,1)$ and $(0,0,1,1)$ which all have pairwise Hamming distance at least $2$. So for these values the answer is in fact $4$ which is as high as it can be.

Set $n = 7$, $m = 4$ and $d=3$. Over all vectors $V$ of length $7$, the largest set $S$ of subvectors of length $4$ all of which have pairwise distance $3$ from each other is $2$.

Set $n=9$, $m=5$ and $d=3$. Vector $V=(0, 0, 0, 1, 1, 0, 1, 0, 0)$ gives you the answer $4$ and is the maximum possible for these values of $n,m,d$.

Set $n=11$, $m=6$ and $d=3$. Vector $V=(0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1)$ gives you the answer $6$ which is as high as it can be.

Set $n=13$, $m=7$ and $d=4$. Vector $V=(0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1)$gives you the answer $7$ which is as high as it can be.

**Clarification II.** I would be happy with bounds rather than exact answers. Is there, for example, an equivalent of the Hamming bound for this setup?

An Introduction to Symbolic Dynamics and Coding, Cambridge 1995. And also that application of Euler paths to recombining fragments of RNA, Tucker "A new applicable proof of the Euler circuit theorem"American Mathematical Monthly83 (1976) 638-640. – Brian Hopkins Apr 28 '13 at 20:42