A probabilistic construction of maximum h-scattered subspaces

Abstract

Let 1≤ h≤ n−2 and g=gcd(n,h+1). We prove that, for every prime power q, a maximum h-scattered 𝔽_q-subspace of V(r,qⁿ) exists whenever (h+1)| rn and r≥(h+1)/g(h(n−h−2)+1). In particular, for fixed n and minimum distance δ∈{2,…,n−1}, 𝔽_qⁿ-linear MRD codes of length N exist for every q and every sufficiently large N such that n divides (n−δ+1)N, with a bound on N depending only on n and δ. We take the 𝔽_q^s-span of the columns of a random matrix A over the algebra R=𝔽_qⁿ⊗_{𝔽_q}𝔽_q^s, for a suitable s. Such a subspace is maximum h-scattered if and only if the kernel of A contains no nonzero vector whose expansion with respect to an 𝔽_q-basis of 𝔽_q^s has rank weight at most h+1. When gcd(n,s)=1, R is a field, and a first-moment argument, counting these vectors up to multiplication by 𝔽_qⁿ^*𝔽_q^s^*, shows that with positive probability the kernel contains none of them if s>h(n−h−2). When d=gcd(n,s)>1, R is a product of d fields, and vectors vanishing in some factors lie in the kernel with larger probability. We control them by proving that, for an (h+1)-dimensional 𝔽_q-subspace D of 𝔽_qⁿ, the elements of D⊗_{𝔽_q}𝔽_q^s vanishing outside k given factors, 0<k<d, form an 𝔽_q^s-space of dimension at most (h+1)k/d, with equality only if D is linear over a subfield of 𝔽_qⁿ larger than 𝔽_q. Taking s=gr/(h+1) gives the bound on r. Finally, if (n−h−1)|(h+1), then for every r satisfying the above inequality there is an h-scattered 𝔽_q-subspace of V(r,qⁿ) of dimension ⌊ rn/(h+1)⌋, without a divisibility condition on rn.

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BibTeX RIS

BibTeX

@misc{giannoni2026probabilistic,
  title = {{A probabilistic construction of maximum $h$-scattered subspaces}},
  author = {Alessandro Giannoni and Alessandro Neri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32737},
  eprint = {2610.32737},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32737},
}

RIS

TY  - UNPB
TI  - A probabilistic construction of maximum h-scattered subspaces
AU  - Alessandro Giannoni
AU  - Alessandro Neri
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32737
UR  - https://arxiv.org/abs/2610.32737
AB  - Let 1≤ h≤ n−2 and g=gcd(n,h+1). We prove that, for every prime power q, a maximum h-scattered 𝔽_q-subspace of V(r,q^n) exists whenever (h+1)| rn and r≥(h+1)/g(h(n−h−2)+1). In particular, for fixed n and minimum distance δ∈{2,…,n−1}, 𝔽_q^n-linear MRD codes of length N exist for every q and every sufficiently large N such that n divides (n−δ+1)N, with a bound on N depending only on n and δ. We take the 𝔽_q^s-span of the columns of a random matrix A over the algebra R=𝔽_q^n⊗_𝔽_q𝔽_q^s, for a suitable s. Such a subspace is maximum h-scattered if and only if the kernel of A contains no nonzero vector whose expansion with respect to an 𝔽_q-basis of 𝔽_q^s has rank weight at most h+1. When gcd(n,s)=1, R is a field, and a first-moment argument, counting these vectors up to multiplication by 𝔽_q^n^*𝔽_q^s^*, shows that with positive probability the kernel contains none of them if s>h(n−h−2). When d=gcd(n,s)>1, R is a product of d fields, and vectors vanishing in some factors lie in the kernel with larger probability. We control them by proving that, for an (h+1)-dimensional 𝔽_q-subspace D of 𝔽_q^n, the elements of D⊗_𝔽_q𝔽_q^s vanishing outside k given factors, 0<k<d, form an 𝔽_q^s-space of dimension at most (h+1)k/d, with equality only if D is linear over a subfield of 𝔽_q^n larger than 𝔽_q. Taking s=gr/(h+1) gives the bound on r. Finally, if (n−h−1)|(h+1), then for every r satisfying the above inequality there is an h-scattered 𝔽_q-subspace of V(r,q^n) of dimension ⌊ rn/(h+1)⌋, without a divisibility condition on rn.
ER  -

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