A probabilistic construction of maximum $h$-scattered subspaces

Alessandro Giannoni; Alessandro Neri

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A probabilistic construction of maximum $h$-scattered subspaces

Alessandro Giannoni, Alessandro Neri

arXiv:2610.32737v1Oct 23, 2026math.COcs.ITmath.IT

TL;DR

The paper addresses when maximum h-scattered subspaces exist, including cases not covered by the divisibility condition for the integer dimension bound. It uses random matrices over an auxiliary field and derives existence results, including long-length MRD-code existence and a dimension-bound result without divisibility on rn.

Problem
The paper asks when maximum h-scattered subspaces exist and, when rn is not divisible by h+1, whether the integer dimension bound can still be attained.
Source: §1 · p. 1
Method
It takes the F_{q^s}-span of random-matrix columns and uses a low-rank kernel criterion to test whether the resulting subspace is maximum h-scattered.
Source: §1 · p. 2
Results
For (n,h)=(8,3), such subspaces exist for every q when r≥10; the paper also proves long-length admissible MRD-code existence with a threshold independent of q.
Source: §1 · p. 2
Takeaways & Limitations
Under Theorem 1.2's parameter condition, the integer dimension floor(rn/(h+1)) is attained without requiring (h+1)|rn.
Source: §1 · p. 3
The direct bound is not sharp, and for h≤n−4 with positive defect it never proves attainment of the integer upper bound.
Source: §5.1 · p. 14

Abstract

from arXiv · show

Let $1\le h\le n-2$ and $g=\gcd(n,h+1)$. We prove that, for every prime power $q$, a maximum $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ exists whenever $(h+1)\mid rn$ and $$r\ge\frac{h+1}{g}\bigl(h(n-h-2)+1\bigr).$$ In particular, for fixed $n$ and minimum distance $\delta\in\{2,\ldots,n-1\}$, $\mathbb{F}_{q^n}$-linear MRD codes of length $N$ exist for every $q$ and every sufficiently large $N$ such that $n$ divides $(n-\delta+1)N$, with a bound on $N$ depending only on $n$ and $\delta$. We take the $\mathbb{F}_{q^s}$-span of the columns of a random matrix $A$ over the algebra $R=\mathbb{F}_{q^n}\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$, 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 $\mathbb{F}_q$-basis of $\mathbb{F}_{q^s}$ 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 $\mathbb{F}_{q^n}^*\mathbb{F}_{q^s}^*$, 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 $\mathbb{F}_q$-subspace $D$ of $\mathbb{F}_{q^n}$, the elements of $D\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$ vanishing outside $k$ given factors, $0<k<d$, form an $\mathbb{F}_{q^s}$-space of dimension at most $(h+1)k/d$, with equality only if $D$ is linear over a subfield of $\mathbb{F}_{q^n}$ larger than $\mathbb{F}_q$. Taking $s=gr/(h+1)$ gives the bound on $r$. Finally, if $(n-h-1)\mid(h+1)$, then for every $r$ satisfying the above inequality there is an $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ of dimension $\lfloor rn/(h+1)\rfloor$, without a divisibility condition on $rn$.

1. Introduction

The introduction frames the existence problem for maximum h-scattered subspaces, presents a random-matrix construction using an auxiliary field, and states existence and coding-theoretic consequences. It also extends the construction to attain the integer dimension bound in a further parameter range.

1. Introduction

Earlier existence results cover (h+1)|r, h=1, h=n−3, and the stated (n,h)=(6,2) cases.When gcd(n,h+1)=1, admissibility forces (h+1)|r, so the direct-sum construction settles existence.Source: §1 · p. 2
The construction takes the F_{q^s}-span of columns of a random matrix and tests failure by whether its expanded matrix has a nonzero kernel vector of rank at most h+1.The auxiliary field degree s is chosen coprime to n for the initial construction.Source: §1 · p. 2
For (n,h)=(8,3), maximum 3-scattered subspaces exist for every q when r≥10, improving on the coprime-degree threshold r≥19.The new argument is needed only when gcd(n,h+1)>1 and 2≤h≤n−4; other cases follow from recalled constructions.Source: §1 · p. 2
For fixed extension degree and minimum distance, all sufficiently large admissible lengths support F_{q^n}-linear MRD codes, with a threshold independent of q.This is presented as an asymptotic form of the long-length case of the linear MRD conjecture [14, Conjecture 4.2(2)].Source: §1 · p. 2
Theorem 1.2 removes the divisibility requirement on rn and gives h-scattered subspaces of dimension floor(rn/(h+1)) under its stated parameter condition.The passage specifies the condition (n−h−1)|(h+1); the theorem's remaining condition is not included in the supplied excerpt.Source: §1 · p. 3
For h=n−2, and for h=n−3 with even n, the integer upper bound is attained for every r≥h+1; for odd n≥7 and h=n−3, further residue classes are obtained.More generally, the procedure gives explicit residue classes of r in which the integer upper bound is attained.Source: §1 · p. 3

2. Low-rank relations and scattered subspaces

This section characterizes h-scattered subspaces through low-rank kernel relations and connects that criterion to MRD codes. It also establishes direct-sum constructions from smaller scattered subspaces.

Low-rank relation criterion

An F_q-span of matrix columns is h-scattered of dimension N exactly when the matrix has row rank r and its kernel has no nonzero vector of rank weight at most h+1.The criterion applies for r≥h+1.Source: §2 · p. 3

Coding-theoretic consequence

When (h+1)|rn, the row space of the matrix is an [N,r,n−h]_{q^n/q} MRD code.The section derives this through the kernel criterion and the Singleton bound.Source: §2 · p. 4

Direct sums

The direct sum of two h-scattered subspaces is h-scattered, and it is maximum when both summands are maximum.The block-diagonal matrix has kernel equal to the direct sum of the two kernels, so a low-rank vector in either block would contradict the criterion.Source: §2 · p. 4

Standard construction

The standard subspace U_st={(x,x^q,…,x^{q^h}):x∈F_{q^n}} is maximum h-scattered in dimension h+1.Its maximum property follows because any h+1 vectors independent over F_q are independent over F_{q^n}, by the Moore determinant.Source: §2 · p. 4

3. A random construction with an auxiliary field

The section constructs candidate subspaces from random matrices over an auxiliary field and characterizes success by excluding low-expanded-rank kernel vectors. For coprime field degrees, a first-moment bound yields an explicit existence condition.

Expanded-rank criterion

A candidate U_A is maximum h-scattered exactly when ker A contains no nonzero vector of expanded rank at most h+1.Expanded rank is computed after expanding coordinates in an F_q-basis of F_{q^s}, and is independent of the chosen basis.Source: §3 · p. 5

Existence bound

When β_{n,h,s}(q)≤1, a random matrix yields a maximum h-scattered subspace with positive probability.The union bound counts projective points whose vectors have expanded rank at most h+1.Source: §3 · pp. 5–6

Coprime-field count

For coprime s and n, the first-moment bound is at most one precisely when s>h(n−h−2).The equality case β=1 occurs only for q=2, s=1, and h=n−2.Source: §3 · p. 6

Coprime-field existence

If gcd(s,n)=1, s>h(n−h−2), (h+1)|nℓ, and sℓ≥h+1, a maximum h-scattered subspace exists that is F_{q^s}-linear.The construction places it in V(sℓ,q^n).Source: §3 · p. 6

4. Auxiliary algebras without coprimality

The auxiliary-algebra argument controls vectors supported on only some factors, enabling existence results beyond coprime auxiliary degrees. Its support bound and equality case feed a uniform threshold and a normalized-dimension construction.

4. Auxiliary algebras without coprimality.

For s with d = gcd(n,s), the algebra R is represented as F_{q^n}[Y]/(m(Y)), where m has d irreducible factors over F_{q^n}.The argument works with the component structure rather than requiring R to be a field.Source: §4 · p. 7
For a proper nonempty support T, dκ_T(D) ≤ (h+1)|T|; equality holds exactly when D is linear over F_{q^{d_T}}.In the strict case, the positive integer gap is divisible by gcd(d,h+1).Source: §4 · pp. 7–9
Theorem 4.1 bounds the contribution from vectors vanishing outside a selected factor set, while full-support vectors retain the coprime-field term.The support inequality is the key control on vectors that otherwise enter the kernel with larger probability.Source: §4 · pp. 7–8

4.2. Existence in every sufficiently large admissible dimension.

For every prime power q, every admissible r ≥ (h+1)/g·(h(n-h-2)+1) admits a maximum h-scattered subspace.The construction uses normalized dimension b = gr/(h+1); if g does not divide b, it applies once b > h(n-h-2).Source: §4.2 · p. 11
When g does not divide b, Lemma 4.3 supplies the required subspace once b > h(n-h-2); when g divides b, standard blocks suffice.Here b = gr/(h+1) is the normalized dimension.Source: §4.2 · p. 11
For (n,h) = (8,3), the étale construction adds normalized dimension 10 to the coprime semigroup, yielding existence for every q and r ≥ 10.The resulting dimensions include r = 4 and 8 as well.Source: §4.2 · p. 12

5. Existence without divisibility

The section extends upper-bound-attaining constructions beyond the divisibility condition, using free coordinates, direct bounds, and additional blocks when h = n −3. The resulting coverage depends on normalized dimension or residue class, with a stated gap for some constructions.

5. Existence without divisibility

Appending j free coordinates to an upper-bound-attaining h-scattered subspace preserves h-scatteredness, and the enlarged subspace attains the integer upper bound exactly when its new defect is less than h + 1.The defect increases by j(n − h − 1).Source: §5 · pp. 12–13
The integer upper bound is attained when r = (h + 1)b/g + j, with b ≥ h(n − h − 2) + 1 and 0 ≤ j ≤ floor(h/(n − h − 1)).When (n − h − 1) divides (h + 1), this yields attainment for every r satisfying the stated bound, without requiring (h + 1) | rn.Source: §5 · p. 13
For h = n −3, the constructions give no information for remaining residues when (h + 1)/g > 1.The values j = 0, …, J are distinct modulo (h + 1)/g.Source: §5 · p. 13

5.1. The direct bound at positive defect.

For h ≤ n −4 and positive defect, the direct probability bound never proves existence at the integer upper bound.The bound proves existence when E_s < 1; its exact cases include defect zero with s > h(n − h − 2), and positive defect for h = n −2.Source: §5.1 · p. 14
The direct probability bound is not sharp because its counting can overcount vectors and the union bound is not exact.Theorem 5.4 classifies when this bound is below one, rather than giving a sharp existence threshold.Source: §5.1 · p. 14

5.2. Additional constructions when h = n −3.

For h = n −3, the additional positive-defect blocks are indexed by residues c > (h + 1)/2 with a nonempty set D_c, and other supplied parameters come from these blocks, standard blocks, and free coordinates.For each such residue, the smallest eligible s_c gives a block for every q.Source: §5.2 · p. 15
When n = h + 3 is odd, the stated constructions attain the integer upper bound for all sufficiently large r exactly in the listed residue classes; when n is even, they attain it for every r ≥ h + 1.The listed odd-n classes comprise 0 through h/2 and the interval represented in the following passage; for h = 2 the second interval is empty.Source: §5.2 · p. 15

Declaration on the use of AI tools

The authors report that Claude (Anthropic, Opus 5.5) assisted with exposition, checking arguments, and developing proofs from author-provided ideas; the authors independently verified the results and proofs and assume responsibility for the manuscript.

Claude assisted with exposition, checking mathematical arguments, and developing proofs based on ideas from the authors; the authors independently verified the results and proofs and assume responsibility for the manuscript.Source: Declaration on the use of AI tools · p. 16