Finite-field adaptation of the rational remote-point algorithm

Adapt the deterministic polynomial-time algorithm for the Remote Point Problem over the rational numbers to finite fields so that it achieves a remoteness parameter d=ω(log n), including particularly the case of the binary field F₂ and, as an intermediate goal, fields whose sizes grow polynomially or exponentially with n.

Background

The paper gives a deterministic polynomial-time algorithm over Q that achieves the optimal remoteness n−k for a k-dimensional subspace. Over finite fields, it only obtains remoteness Ω(n/(max{k, log n}·log n)), so adapting the rational-number technique to achieve superlogarithmic remoteness would substantially improve the finite-field results.

The authors identify small fields, especially F₂, as the likely hardest setting and suggest fields of polynomial or exponential size as possible intermediate cases. Even for exponentially large fields, field elements can still be represented using only polynomially many bits.

References

This work raises several natural open problems.

Can the algorithm from \cref{thm:rpp-over-Q} be adapted to obtain a deterministic algorithm for RPP over finite fields, for any remoteness parameter $d=\omega(\log n)$? Small fields like $F_2$ seem like the hardest case; perhaps a more viable intermediate goal is to consider fields with size growing polynomially or even exponentially with $n$, noting that even when the field size is exponential in $n$, representing each element in it requires only polynomially many bits.

— Improved Algorithms for the Remote Point Problem  (2609.25765 - Volk, 22 Sep 2026) in Section Open Problems, first item