Maximum 2-scattered subspaces of V(r,q^6) for every r≥ 3

Abstract

We prove that for every prime power q and every integer r≥ 3 there exists a maximum $2$-scattered 𝔽_q-subspace of V(r,q⁶), that is, a 2r-dimensional 𝔽_q-subspace meeting every $2$-dimensional 𝔽_q⁶-subspace in dimension at most $2$. Since an example in dimension three is known and direct sums preserve the property, it suffices to treat r=4 and r=5. For r=4 we study the subspaces U_b={(x,x^q+bx^q⁵):Tr_q¹²/q⁴(x)=0} of 𝔽_q¹²², b∈𝔽_q¹². We describe the parameters b for which U_b is not maximum $2$-scattered in terms of trace conditions and Dickson determinants, and deduce that at least 1/2q¹²−q¹¹+O(q¹⁰) parameters work; this number is positive for q≥ 3, and a computer search settles q=2. For every r≥ 5 coprime to $6$ we show that U_s={x+ε s x^{q^r}:x∈𝔽_q^2r}⊆𝔽_q^6r is maximum $2$-scattered for all ε∈𝔽_q⁶∖𝔽_q² and s∈𝔽_q^r∖𝔽_q. Consequently, 𝔽_q⁶-linear [2r,r,4]_q⁶/q MRD codes exist for every r≥ 3; for r coprime to $6$, the codes obtained from U_s are equivalent to their duals.

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BibTeX RIS

BibTeX

@misc{bartoli2026maximum,
  title = {{Maximum 2-scattered subspaces of $V(r,q^6)$ for every $r\geq 3$}},
  author = {Daniele Bartoli and Francesco Ghiandoni and Alessandro Giannoni and Giuseppe Marino and Alessandro Neri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08237},
  eprint = {2610.08237},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08237},
}

RIS

TY  - UNPB
TI  - Maximum 2-scattered subspaces of V(r,q^6) for every r≥ 3
AU  - Daniele Bartoli
AU  - Francesco Ghiandoni
AU  - Alessandro Giannoni
AU  - Giuseppe Marino
AU  - Alessandro Neri
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.08237
UR  - https://arxiv.org/abs/2610.08237
AB  - We prove that for every prime power q and every integer r≥ 3 there exists a maximum $2$-scattered 𝔽_q-subspace of V(r,q^6), that is, a 2r-dimensional 𝔽_q-subspace meeting every $2$-dimensional 𝔽_q^6-subspace in dimension at most $2$. Since an example in dimension three is known and direct sums preserve the property, it suffices to treat r=4 and r=5. For r=4 we study the subspaces U_b={(x,x^q+bx^q^5):Tr_q^12/q^4(x)=0} of 𝔽_q^12^2, b∈𝔽_q^12. We describe the parameters b for which U_b is not maximum $2$-scattered in terms of trace conditions and Dickson determinants, and deduce that at least 1/2q^12−q^11+O(q^10) parameters work; this number is positive for q≥ 3, and a computer search settles q=2. For every r≥ 5 coprime to $6$ we show that U_s={x+ε s x^q^r:x∈𝔽_q^2r}⊆𝔽_q^6r is maximum $2$-scattered for all ε∈𝔽_q^6∖𝔽_q^2 and s∈𝔽_q^r∖𝔽_q. Consequently, 𝔽_q^6-linear [2r,r,4]_q^6/q MRD codes exist for every r≥ 3; for r coprime to $6$, the codes obtained from U_s are equivalent to their duals.
ER  -

Maximum 2-scattered subspaces of V(r,q⁶) for every r≥ 3

Abstract

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