PARTIAL SPREADS AND VECTOR SPACE PARTITIONS
THOMAS HONOLD, MICHAEL KIERMAIER, AND SASCHA KURZ
Abstract. Constant dimension codes with the maximum possible minimum
distance have been studied under the name of partial spreads in finite geometry
for several decades. It is no surprise that the sharpest bounds on the maximal
code sizes are typically known for this subclass. The seminal works of Andr´e,
Segre, Beutelspacher, and Drake & Freeman date back to 1954, 1964, 1975, and
1979, respectively. Until recently, there was almost no progress besides some
computer based constructions and classifications. It turns out that vector space
partitions provide the appropriate theoretical framework, Here, we provide an
historic account and an interpretation of the classical results from a modern
point of view. To this end, we introduce all required methods from the theory
of vector space partitions and finite geometry in a tutorial style. We guide the
reader to the current frontiers of research in that field.
1. Introduction
Let F
q
be the finite field with q elements, where q > 1 is a prime power. By F
v
q
we denote the standard vector space of dimension v ≥ 1 over F
q
. The set of all
subspaces of F
n
q
, ordered by the incidence relation ⊆, is called (v −1)-dimensional
projective geometry over F
q
and denoted by PG(v −1, F
q
). It forms a finite modular
geometric lattice with meet X ∧Y = X ∩Y and join X ∨Y = X + Y . The points of
PG(v−1, F
q
) are the 1-dimensional subspaces of F
v
q
. Instead of PG(v−1, F
q
) we will
mainly use the notation F
v
q
in the following, pointing to fact that the dimensions
differ by one in both notations. The set of all k-dimensional subspaces of an F
q
-
vector space V will be denoted by
V
k
q
. For v = dim(V ), its cardinality is given by
the Gaussian binomial coefficient
v
k
q
:=
(
(q
v
−1)(q
v−1
−1)···(q
v−k+1
−1)
(q
k
−1)(q
k−1
−1)···(q−1)
if 0 ≤ k ≤ v;
0 otherwise,
which does not depend on the precise choice of the F
q
-vector space V . The latter
fact is true for all of the remaining definitions of this chapter, so that we will stick to
the specialized notation F
v
q
. Two widely used metrics are are given by the subspace
distance d
S
(X, Y ) := dim(X + Y ) − dim(X ∩ Y ) = 2 · dim(X + Y ) − dim(X) −
dim(Y ), which is indeed the graph theoretic distance in the geometric lattice, and
the injection distance d
I
(X, Y ) := max {dim(X), dim(Y )}−dim(X ∩Y ), where X
and Y are subspaces of F
v
q
. A set C of subspaces of F
v
q
is called a subspace code.
The minimum (subspace) distance of C is given by d = min{d
S
(X, Y ) | X, Y ∈
C, X 6= Y }. If all elements of a subspace code C have the same dimension, say
k, we speak of a constant dimension code. For a constant dimension code C we
have d
S
(X, Y ) = 2d
I
(X, Y ) for all X, Y ∈ C, so that we will restrict ourselves to
the subspace distance in the following. An important problem is the determination
of the maximum possible cardinality A
S
q
(v, d; k) of a constant dimension code with
minimum subspace distance d in F
v
q
, where all codewords have dimension k. For two
codewords X and Y of dimension k a subspace distance of at least d corresponds
to dim(X ∩ Y ) ≤
2k−d
2
. Thus, the maximum possible minimum distance of a
constant dimension code with codewords of dimension k is 2k. This extremal case
1
2 THOMAS HONOLD, MICHAEL KIERMAIER, AND SASCHA KURZ
was also studied under the name of partial k-spreads in finite geometry for several
decades, i.e., partial k-spreads are collections of k-dimensional subspaces of F
v
q
with
trivial, i.e., zero-dimensional intersection. If the all-zero vector 0 is removed from
the codewords, then the resulting objects can by packed into F
v
q
\0. In terms of
the projective geometry partial spreads are packings of the set of points. It is no
surprise that the sharpest bounds on the maximal code sizes A
S
q
(v, d; k) of constant
dimension codes are typically known for this special subclass of partial spreads. In
the case of a perfect packing, i.e., a partition, we speak of a k-spread. Partitions
of F
n
q
\0, or the set of points of PG(v − 1, F
q
), with elements of possibly different
dimensions have been introduced under the name vector space partitions. A vector
space partition P of F
v
q
is a collection of subspaces with the property that every
non-zero vector is contained in a unique member of P. If P contains m
d
subspaces
of dimension d, then P is of type k
m
k
. . . 1
m
1
. We may leave out some of the cases
with m
d
= 0. So, partial k-spreads are just a special case of vector space partitions,
where the elements all have a dimension of either k or 1. The elements of dimension
1 arise as the set of uncovered elements of points, that is why they are called holes
in this context. It turns out that vector space partitions provide an appropriate
framework to study bounds on the sizes of partial spreads.
There is a huge amount of related work that we will not cover. Partial spreads
have also been studied for designs and in polar spaces, see e.g. [2, 16]. For the special
case v = 2k there is a connection to translation planes [38]. From a geometric point
of view, several researchers are interested in (inclusion) maximal partial spreads,
while we consider only those of maximal cardinality. The classification, see e.g. [41],
of all partial spreads up to isomorphism is also not treated. There is a stream of
literature that characterizes the existing types of vector space partitions in F
v
2
for small dimensions v. Here, we give only partial details for results that are
independent of the dimension v of the ambient space and refer to [23] otherwise.
The remaining part of this chapter is structured as follows. In Section 2 we review
some, mostly classical, bounds and constructions for partial spreads. Introducing
the concept of q
r
-divisible sets and codes is introduce in Section 3, we are able to
obtain improved upper bounds for partial spreads in Theorem 9 and Theorem 10.
Constructions for q
r
-divisible sets are presented in Section 4. Some more non-
existence results for q
r
-divisible sets are presented in Section 5 before we end with
a collection of open research problem in Section 6.
2. Bounds and constructions for partial spreads
Counting the points in F
v
q
and F
k
q
gives the obvious upper bound A
q
(v, 2k; k) ≤
v
1
q
/
k
1
q
= (q
v
− 1) /
q
k
− 1
. The case of equality corresponds to the situation
of spreads. Essentially, based on the idea of the Andr´e-Bruck-Bose construction
[1, 10, 38], i.e., the connection to translation planes, there is the following complete
characterization by Segre from 1964.
Theorem 1. ([46], [13, p. 29]) F
v
q
contains a k-spread if and only if k divides v,
where we assume 1 ≤ k ≤ v and k, v ∈ N.
Since
q
v
−1
q
k
−1
is an integer if and only if k divides v only the constructive part
needs to be shown. In this case a k-spread can be constructed from the so-called
subfield construction or field reduction, see e.g. [39] for an extensive review. To
this end write v = kt for a suitable integer t. The
t
1
q
k
=
q
v
−1
q
k
−1
points, i.e., the 1-
dimensional q
k
-subspaces, of F
t
q
k
clearly form a partition. Each such point consists
of q
k
−1 non-zero vectors. Considering F
q
k
as a k-dimensional q-vector space maps
the points of F
t
q
k
to k-dimensional subspaces of F
v
q
with trivial intersection. As an
PARTIAL SPREADS AND VECTOR SPACE PARTITIONS 3
example we consider the parameters q = 3, v = 4, and k = 2. Using canonical
representatives in F
9
' F
3
[x]/(x
2
+ 1) the
2
1
9
= 10 points in F
2
9
are generated by
0
1
,
1
0
,
1
1
,
1
2
,
1
x
,
1
x + 1
,
1
x + 2
,
1
2x
,
1
2x + 1
,
1
2x + 2
.
The point
1
x + 1
· F
9
maps to the nine vectors
0
0
0
0
,
1
0
1
1
,
2
0
2
2
,
0
1
2
1
,
0
2
1
2
,
1
1
0
2
,
2
2
0
1
,
2
1
1
0
,
1
2
2
0
,
which form a 2-dimensional subspace in F
4
3
.
If k does not divide v, then we can improve the stated upper bound by rounding
down to A
q
(v, 2k; k) ≤
j
q
v
−1
q
k
−1
k
, since A
q
(v, 2k; k) obviously is an integer. How-
ever, this bound can be improved further. Before we go into the details we present
another construction – the so-called Echelon-Ferrers construction for general sub-
space codes, see [18]. To this end, we remark that each k-dimensional subspace
of F
v
q
can be written as the row-space of a full rank matrix A ∈ F
k×v
q
. Since the
application of the Gaussian elimination algorithm onto a matrix A does not change
its row-space, the resulting matrix in reduced row echelon form (rre) can be used
as a canonical representative for each subspace. To be precise, we denote the bijec-
tion from the subspaces to the representing matrices by τ . In our example the two
vectors
1
x + 1
·1,
1
x + 1
·x ∈ F
2
9
form a F
3
-basis. Mapped to F
4
3
and written in
rows, we obtain the (generator) matrix A =
1 0 1 1
0 1 2 1
, which is already in rre.
Given a full rank matrix A ∈ F
k×v
q
, we denote by p(A) ∈ F
v
2
the binary vector whose
1-entries coincide with the pivot columns of A. For each π ∈ F
v
2
let EF
q
(π) denote
the set of all k ×v matrices over F
q
that are in reduced row echelon form with pivot
columns described by π, where k is the (Hamming) weight of π. In our example we
have π = p(A) = (1, 1, 0, 0) and, more generally, EF
q
(π) =
1 0 ? ?
0 1 ? ?
, where
the ?s represent arbitrary elements of F
q
, i.e., |EF
q
(v)| = q
4
. In general we have
EF
q
(π
1
, . . . , π
n
)
= q
n
P
i=1
(1−π
i
)·
i
P
j=1
π
j
and the structure of the corresponding ma-
trices can be read off from the corresponding (Echelon)-Ferrers diagram
1
• •
• •
,
where the pivot columns and zeros are omitted and the stars are replaced by black
disks.
For matrices A, B ∈ F
m×v
q
the rank distance is defined via d
R
(A, B) := rk(A−B).
The subspace distance of two subspaces with the same pivots can be computed by
the rank distance of the corresponding generator matrices, see e.g. [47, Corollary
3].
Lemma 1. Let π ∈ F
v
2
and X, Y ∈ EF
q
(π), then d
S
(X, Y ) = 2 · d
R
τ(X), τ (Y )
.
Let d
H
(π, π
0
) := |{1 ≤ i ≤ n : π
i
6= π
0
i
}| denote the Hamming distance for two
binary vectors π, π
0
∈ F
v
2
. The subspace distance of two subspaces with different
1
A Ferrers diagram represents partitions as patterns of dots, with the ith row having the same
number of dots as the ith term s
i
in the partition v = s
1
+ · · · + s
l
, where s
1
≥ · · · ≥ s
l
and
s
i
∈ N
>0
.
4 THOMAS HONOLD, MICHAEL KIERMAIER, AND SASCHA KURZ
pivots can be upper bounded via the Hamming distance of the corresponding pivot
vectors, see [18, Lemma 2].
Lemma 2. Let π, π
0
∈ F
v
2
, X ∈ EF
q
(π), and Y ∈ EF
q
(π
0
), then d
S
(X, Y ) ≥
d
H
(π, π
0
).
Having Lemma 1 and Lemma 2 at hand, the Echelon-Ferrers construction from
[18] works as follows: Choose a binary code S of length v and minimum Hamming
distance 2δ as a so-called skeleton code. For each s ∈ S construct a code C
s
⊆ EF
q
(s)
having a minimum rank distance of at least δ. Setting C = ∪
s∈S
C
s
yields a subspace
code whose minimum distance is at least 2δ. The weights of the pivot vectors in S
correspond to the dimensions of the codewords in C.
For a vector π ∈ F
v
2
and an integer 1 ≤ δ ≤ v let q
dim(π,δ)
be the largest
cardinality of a linear rank-metric code over EF
q
(π) with rank distance at least δ.
Theorem 2. ([18, Theorem 1]) For a given i, 0 ≤ i ≤ δ −1, if ν
i
is the number of
dots in the Echelon-Ferrers diagram corresponding to π, which are not contained in
the first i rows and not contained in the rightmost δ −1 −i columns, then min
i
{ν
i
}
is an upper bound of dim(π, δ).
The conjecture that the upper bound of Theorem 2 can be obtained for all
parameters is still open. A special subcase is given by rectangular Ferrers diagrams.
Theorem 3. ([19]) Let m, v ≥ d be positive integers and C ⊆ F
m×v
q
be a rank-
metric code with minimum rank distance d. Then, |C| ≤ q
max(v,m)·(min(v,m)−d+1)
.
Codes attaining this upper bound are called maximum rank distance (MRD) codes.
They exist for all (suitable) choices of parameters.
If m < d or v < d, then only |C| = 1 is possible, which may be summarized to
the single upper bound |C| ≤
q
max(v,m)·(min(v,m)−d+1)
. Using an m × m identity
matrix as a prefix one obtains the so-called lifted MRD codes.
Theorem 4. ([48]) For k, d, v ∈ N
>0
with k ≤ v, d ≤ 2 min(k, v − k), d even,
the size of a lifted MRD code with subspace distance d is given by M (q, k, v, d) :=
q
max(k,v−k)·(min(k,v−k)−d/2+1)
. If d > 2 min(k, v−k), then we have M(q, k, v, d) = 1.
So, taking binary vectors π
i
, where the ones are located in positions (i −1)k + 1
to ik, for all 1 ≤ i ≤ bv/kc, clearly gives a binary constant weight code of length v,
weight k, and minimum Hamming distance 2k.
1 . . . 10 . . . 00 . . . 00 . . .
0 . . . 01 . . . 10 . . . 00 . . .
0 . . . 00 . . . 01 . . . 10 . . .
. . .
Lemma 3. For positive integers k, v with v > 2k and v 6≡ 0 (mod k), there exists
a partial k-spread in F
n
q
having cardinality
1+
bv/kc−1
X
i=1
q
v−ik
= 1+q
k+(v mod k)
·
q
v−k−(v mod k)
− 1
q
k
− 1
=
q
v
− q
k+(v mod k)
+ q
k
− 1
q
k
− 1
.
Given the (v mod k)-term, a specific parameterization is useful: Write v = kt+r,
where 1 ≤ r ≤ k − 1, and A
q
(kt + r, 2k; k) = q
r
·
q
kt
−1
q
k
−1
− s. Lemma 3 gives
s ≤ q
r
−1 and there was the wrong conjecture that this bound is sharp. s ≥ q − 1
and s >
q
r
−1
2
−
q
2r−k
5
is known, see e.g. [16] and the details presented later on.
Note that v ≡ r (mod k), so that the residue class r seems to play a major role.
Besides the case of r = 0, see Theorem 1, the next case r = 1 is solved in full
generality in 1975 by Beutelspacher:
PARTIAL SPREADS AND VECTOR SPACE PARTITIONS 5
Theorem 5. ([4]; see also [27] for the special case q = 2) For integers 1 ≤ k ≤ v
with v ≡ 1 (mod k) we have A
q
(v, 2k; k) = q
1
·
q
v−1
−1
q
k
−1
− q + 1 =
q
v
−q
k+1
+q
k
−1
q
k
−1
.
Proof. Let C be a partial k-spread in F
kt+1
q
of cardinality q ·
q
kt
−1
q
k
−1
− s, where
s ≤ q − 1. Since each codeword is contained in
k(t−1)+1
1
q
hyperplanes and the
number of hyperplanes is given by
kt+1
1
q
, the average number of codewords per
hyperplane is
|C|·
[
k(t−1)+1
1
]
q
[
kt+1
1
]
q
> q·
q
k(t−1)
−1
q
k
−1
. Thus, there exists a hyperplane containing
at least α := q ·
q
k(t−1)
−1
q
k
−1
+ 1 codewords. Since α ·
k
1
q
+ (|C|− α) ·
k−1
1
q
≤
kt
1
q
,
we have |C| ≤ q ·
q
kt
−1
q
k
−1
− (q −1).
In his original proof Beutelspacher considered the set of holes N and the average
number of holes per hyperplane, which is less than the total number of holes divided
by q. The crucial insight was the relation |N| ≡ |H ∩ N| (mod q
k−1
) for each
hyperplane H, i.e., the number of holes per hyperplane satisfies a certain modulo
constraint. We will see this concept in full generality in Section 3. In terms of
integer linear programming, the upper bound is obtained by an integer rounding
cut. The construction in [4, Theorem 4.2] recursively uses arbitrary k
0
-spreads, so
that it is more general than the one of Lemma 3.
Corollary 1. A
2
(2m, 4; 2) =
2
2m
−1
3
and A
2
(2m + 1, 4; 2) =
2
2m+1
−5
3
for all m ∈
N
≥2
.
For a long time the best upper bound on A
q
(v, 2k; k), i.e., the best known lower
bound on s was the one obtained by Drake and Freeman in 1979:
Theorem 6. (Corollary 8 in [15]) If v = kt + r with 0 < r < k, then
A
q
(v, 2k; k) ≤
t−1
X
i=0
q
ik+r
− bθc − 1 = q
r
·
q
kt
− 1
q
k
− 1
− bθc − 1,
where 2θ =
p
1 + 4q
k
(q
k
− q
r
) − (2q
k
− 2q
r
+ 1).
The authors concluded from the existence of a partial spread the existence of
a (group constructible) (s, r, µ)-net and applied [8, Theorem 1B] – a necessary
existence criterion formulated for orthogonal arrays of strength 2 by Bose and Bush
in 1952. The underlying proof technique can be further traced back to [45] and is
strongly related to the classical second-order Bonferroni Inequality [7, 20], see e.g.
[29, Section 2.5] for another application for bounds on subspace codes.
Given Theorem 1 and Theorem 5 the first open binary case is A
2
(8, 6; 3). The
construction from Lemma 3 gives a partial spread of cardinality 33, while Theorem 6
implies an upper bound of 34. In 2010 the authors of [17] found a sporadic partial
3-spread of cardinality 34 by a computer search. This completely answers the
situation for partial 3-spreads in F
n
2
as shown by the following easy lemma.
Lemma 4. If A
q
(kt
0
+ r, 2k; k) ≥ q
r
·
q
kt
0
−1
q
k
−1
− s for some integer s, then A
q
(kt +
r, 2k; k) ≥ q
r
·
q
kt
−1
q
k
−1
− s for all t ≥ t
0
.
Proof. Let C be a partial k-spread of cardinality q
r
·
q
kt
0
−1
q
k
−1
−s in F
kt
0
+r
q
. Embed C
into GF
kt+r
q
such that the non-zero entries of the pivot vectors of the corresponding
codewords are contained in the last kt
0
+ r coordinates. So, we can apply the
construction of Lemma 3 using only the first k(t − t
0
) coordinates and append the
embed code C, see Lemma 2.
