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In the context of my research, I have to work with sets of vectors $\left\{y_i\right\}_{i\in[n+1]}$ of $\mathbb{F}_2^n$ such that the following property is true: $$\forall i\in[n+1], \left\{y_i\oplus y_j\right\}_{j\in[n+1]\setminus\{i\}}\text{ is a basis of }\mathbb{F}_2^n.$$ Note that we see here $\mathbb{F}_2^n$ as an $\mathbb{F}_2$-vector space with the addition being the bitwise XOR, just like when seeing it as a finite field. Its dimension is thus $n$.

More generally, in order to generate them in some programming language, the definition can be a bit generalized to "is a linearly independent family of vectors", if the set in question has less than $n+1$ elements.

Is this some well-known object, or a particular case of a well-known object?

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    $\begingroup$ I think $n+1$ such vectors are said to be "in general position". $\endgroup$
    – Andreas Blass
    Commented Apr 30 at 15:13
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    $\begingroup$ You're right @AndreasBlass . It's called in general linear position, where usually the basis is $\{y_i - y_j\}_{j \neq i}$ $\endgroup$
    – Maarten Havinga
    Commented May 1 at 11:10
  • $\begingroup$ @AndreasBlass Thanks, this is what I was looking for! Can you add it as an answer so that I can accept it please? $\endgroup$
    – Tristan Nemoz
    Commented May 2 at 8:22
  • $\begingroup$ Because this about $\Bbb{F}_2$ in particular I think your condition is equivalent to the original set of $n+1$ vectors being affinely independent.. $\endgroup$
    – Jyrki Lahtonen
    Commented May 3 at 14:49

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