Heden's bound on the tail of a vector space partition

SASCHA KURZ^?

ABSTRACT. A vector space partition ofF^vq is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring dimension in a vector space partition. To this end, we introduce the notion ofq^r-divisible sets ofk-subspaces inF^vq. By geometric arguments we obtain non-existence results for these objects, which then imply the improved result of Heden.

1. INTRODUCTION

Letq >1be a prime power,Fqbe the finite filed withqelements, andva positive integer. Avector space partitionP ofF^vq is a collection of subspaces with the property that every non-zero vector is contained in a unique member ofP.

IfP containsm_dsubspaces of dimensiond, thenP is of typek^mk. . .1^m1. We may leave out some of the cases with md = 0. Subspaces of dimensiondare also calledd-subspaces.1-subspaces are calledpoints,(v−1)-subspaces are calledhyperplanes, and eachk-subspace containsk

1

q := ^q_q−1^k−1 points. So, in a vector space partitionPeach point of the ambient spaceF^vq is covered by exactly one point of one of the elements ofP. An example of a vector space partition is given by ak-spreadinF^vq, wherev

1

q/k 1

qk-subspaces partition the set of points ofF^vq. The corresponding type is given byk^mk, wheremk = v

1

q/k 1

q. Ifkdividesv then considering the points ofF^v/k_qk as k-dimensional subspaces overFq gives a construction of k-spreads. Ifkdoes not dividev, then nok-spreads exist. Vector space partitions of typek^mk1^m1 are known under the namepartialk-spreads. More precisely, a partialk-spread inF^vq is a setKofk-subspaces such that each point of the ambient spaceF^vq is covered at most by one of its elements. Adding the set of uncovered points, which are also calledholes, gives a vector space partition of typek^mk1^m1. Maximizing m_k = #Kis equivalent to the minimization ofm₁. Ifd₁is the smallest dimension withm_d1 6= 0, we callm_d1 the length of the tailand call the set of the correspondingd₁-subspace thetail. Vector space partitions with a tail of small length are of special interest. In [4] Olof Heden obtained:

Theorem 1. (Theorem 1 in[4]) LetPbe a vector space partition of typedlu_l

. . . d2u₂

d1u₁

ofF^vq, whereu1, u2 >0 anddl>· · ·> d2> d1≥1.

(i) Ifq^d2−d1 does not divideu1and ifd2<2d1, thenu1≥q^d1+ 1;

(ii) ifq^d2−d1does not divideu₁and ifd₂≥2d₁, then eitherd₁dividesd₂andu₁=d₂ 1

q/d₁ 1

qoru₁>2q^d2−d1; (iii) ifq^d2−d1dividesu1andd2<2d1, thenu1≥q^d2−q^d1+q^d2−d1;

(iv) ifq^d2−d1dividesu1andd2≥2d1, thenu1≥q^d2.

Moreover, in Theorem 2 and Theorem 3 he classified the possible sets of d1-subspaces foru1 = q^d1 + 1and u1=d₂

1

q/d₁ 1

q, respectively. The results were obtained using the theory of mixed perfect1-codes, see e.g. [6].

In [2] the authors improved a lower bound of Heden on the size of inclusion-maximal partial2-spreads by translating the underlying techniques into geometry. Here we improve Theorem 1(ii). The underlying geometric structure is the setN ofd₁-subspaces of a vector space partitionPof typed_l^ul. . . d₂^u2d₁^u1. Ford₁this is just a set of points inF^vq. It can be shown that the existence ofPimplies#N ≡# (N ∩H) (mod q^d2−1)for every hyperplaneHofF^vq, see e.g.

[7]. Taking a vector representation of the elements ofN as columns of a generator matrix, we obtain a corresponding (projective) linear codeC overFq. The modulo constraints forN are equivalent to the property that the Hamming weights of the codewords ofC are divisible byq^d2−1. The study of so-called divisible codes, where the Hamming weights of the codewords of a linear code are divisible by some factor∆>1, was initiated by Harold Ward, see e.g.

[9]. The MacWilliams identities, linking the weight distribution of a linear code with the weight distribution of its dual code, can be relaxed to a linear program. Incorporating some information about the weight distribution of a linear code may result in an infeasible linear program, which then certifies the non-existence of such a code. This technique is known under the name linear programming method for codes and was more generally developed for association schemes by Philip Delsarte [3]. In [8] analytic solutions of linear programs for projectiveq^r-divisible linear codes have been applied in order to compute upper bounds for partialk-spreads. Indeed, all currently known upper bounds for partialk-spreads can be deduced from this method, see [7] for a survey.

?Grant KU 2430/3-1 –Integer Linear Programming Models for Subspace Codes and Finite Geometry– German Research Foundation.

1

Here, we generalize the approach to the cased1 >1by studying the properties of the setN ofd1-subspaces of a vector space partitionP ofF^vq of typedlu_l

. . . d2u₂

d1u₁

in Section 2. It turns out that we have#N ≡ #(N ∩H) (modq^d2−d1)for every hyperplaneH ofF^vq, see Lemma 3, which we introduce as a definition of aq^d2−d1-divisible set ofk-subspaces with trivial intersection. By elementary counting techniques we obtain a partial substitute for the MacWilliams identities, see the equations (1) and (2). These imply some analytical criteria for the non-existence of such setsN, which are used in Section 3 to reprove Theorem 1. By an improved analysis we tighten Theorem 1 to Theorem 12. More precisely, the second lower bound of Theorem 1(ii) is improved. We close with some numerical results on the spectrum of the possible cardinalities ofN and pose some open problems.

2. SETS OF DISJOINTk-SUBSPACES AND THEIR INCIDENCES WITH HYPERPLANES

For a positive integerkletN be a set of pairwise disjoint, i.e., having trivial intersection,k-subspaces inF^vq, where we assume that thek-subspaces fromN spanF^vq, i.e.,vis minimally chosen. Bya_iwe denote the number of hyperplanes HofF^vq with#(N ∩H) := #{U ∈ N : U ≤H}=iand setn:= #N. Due to our assumption on the minimality of the dimensionvnot allnelements fromN can be contained in a hyperplane. Double-counting the incidences of the tuples(H),(B1, H), and(B1, B2, H), whereHis a hyperplane andB16=B2are elements ofN contained inH gives:

n−1

X

i=0

ai= v

1

q

,

n−1

X

i=0

iai=n· v−k

1

q

, and

n−1

X

i=0

i(i−1)ai=n(n−1)·

v−2k 1

q

. (1)

For three different elementsB1, B2, B3ofN their spanhB1, B2, B3ihas a dimensionibetween2kand3k. Denoting the number of corresponding triples bybi, double-counting tuples(B1, B2, B3, H), whereH is a hyperplane and B₁, B₂, B₃are pairwise different elements ofN contained inH, gives:

n−1

X

i=0

i(i−1)(i−2)ai=

3k

X

i=2k

bi

v−i 1

q

and

3k

X

i=2k

bi=n(n−1)(n−2). (2) Given parametersq,k,n, andvthe so-called(integer) linear programming methodasks for a solution of the equation system given by (1) and (2) withai, bi ∈R≥0(ai, bi ∈ N). If no solution exists, then no corresponding setN can exist. Fork= 1the equations from (1) and (2) correspond to the first four MacWilliams identities, see e.g. [7].

If there is a single non-zero valueaithe system can be solved analytically.

Lemma 2. Ifa_i= 0for alli6=r >0andk < vin the above setting, then there exists an integers≥2withv=sk andN is ak-spread. Additionally we haver= ^qv−k_qk−1⁻¹.

PROOF. Solving (1) forr,ar, andngivesn= ^q2v−k_qv−q^{−qv−kv−q}−q^v−kk+1⁺¹. Writingv=sk+twiths, t∈Nand0≤t < kwe obtainn=Ps

i=1q^v−ik+^qv−k+t_qv−q^{−qv−kv−k}−q^−qk+1^t+1. Sincen∈Nand0≤q^v−k+t−q^v−k−q^t+ 1< q^v−q^v−k−q^k+ 1 we haveq^v−k+t−q^v−k−q^t+ 1 = 0so thatt= 0andn=^q_q^v_k⁻¹₋₁. Counting points gives thatN partitionsF^vq. We remark thatr= 0forcesn∈ {0,1}so thatN is empty or consists of a singlek-subspace inF^kq andv=kimplies the latter case. So, these degenerated cases correspond tos∈ {0,1}in Lemma 2. As pointed out after [4, Theorem 2], such results can be proved in different ways. While the case that only oneaiis non-zero is rather special, we can show that manyaiare equal to zero in our setting.

Lemma 3. LetP be a vector space partition of typedlu_l

. . . d2u₂

d1u₁

ofF^vq, whereu1, u2>0, and letN be the set ofd1-subspaces. Then, we have#N ≡#(N ∩H) (modq^d2−d1)for every hyperplaneH ofF^vq.

PROOF. For eachU ∈ Pwe havedim(U∩H)∈ {dim(U),dim(U)−1}. So counting points inF^vq andH gives the existence of integersa, a⁰withm·d2

1

q+aq^d2+u1d1

1

q =v 1

qandm·d2−1 1

q+a⁰q^d2−1+u⁰₁q^d1−1+u1d1−1 1

q = v−1

1

q, wherem:=Pl

i=2u_iandu⁰₁:= #(N ∩H). By subtraction we obtainmq^d2−1+aq^d2−a⁰q^d2−1+u₁q^d1−1− u⁰₁q^d1−1=q^v−1, so thatu1q^d1−1≡u⁰₁q^d1−1 (mod q^d2−1).

Definition 4. LetN be a set ofk-subspaces inF^vq. If there exists a positive integerrsuch thata_iis non-zero only if

#N −iis divisible byq^rand thek-subspaces are pairwise disjoint, then we callN q^r-divisible.

Using the notation of Lemma 3,N isq^d2−d1-divisible. As mentioned in the introduction, ford₁ = 1, taking the elements ofN as columns of a generator matrix, we obtain a projective linear code, whose Hamming weights are divisible byq^d2−1.

Example 5. For integersk ≥ 2andr = ak+bwith0 ≤ b < kletN be ak-spread ofF^(a+2)kq . Starting from a(a+ 2)k-spread inF^2(a+2)kq we obtain a vector space partitionP by replacing one(a+ 2)k-dimensional spread

element withN. From Lemma 3 andq^r|q^(a+2)k−k=q^(a+1)kwe deduce that the setN ofk-subspaces isq^r-divisible.

Its cardinality is given by(a+2)k 1

q/k 1

q.

Example 6. For integersk ≥ 2andr ≥ 1letn = k+rand consider a matrix representationM: Fqⁿ → F^n×nq

of Fqⁿ/Fq, obtained by expressing the multiplication maps µα: Fqⁿ → Fqⁿ, x 7→ αx, which are linear overFq, in terms of a fixed basis of Fqⁿ/Fq. Then, all matrices in M(Fqⁿ)are invertible and have mutual rank distance dR(A, B) := rk(A−B) =n, see e.g. [7] for proofs of these and the subsequent facts. In other words, the matrices of M(Fqⁿ)form a maximum rank distance code with minimum rank distancenand cardinalityqⁿ.

Now letB ⊆F^k×nq be the matrix code obtained fromM(Fqⁿ)by deleting the lastn−krows, say, of every matrix.

ThenBhas cardinality minimum rank distancek. Hence, by applying the lifting constructionB7→(I_k|B), whereI_k is thek×kidentity matrix, toBwe obtain a partialk-spreadN inF^vq of sizeqⁿ =q^k+r. Since precisely the points outside the (k+r)-subspaceS=

x∈F^vq : x₁=x₂=· · ·=x_k = 0 are covered,P =N ∪ {S}is a vector space partition ofF^2k+rq andN isq^k+r-divisible with cardinalityq^k+r.

From the first two equations of (1) we deduce:

Lemma 7. For aq^r-divisible setN ofk-subspaces inF^vq, there exists a hyperplaneHwith#(N ∩H)≤n/q^k. PROOF. Letibe the smallest index withai6= 0. Then, the first two equations of (1) are equivalent toP

j≥0ai+q^rj= v

1

q andP

j≥0(i+q^rj)·a_i+qrj =nv−k 1

q. Subtractingitimes the first equation from the second equation gives P

j>0q^rja_i+qrj =n·^qv−k_q−1⁻¹ −i·^q_q−1^v−1. Since the left-hand side is non-negative, we havei≤^qv−k_qv−1⁻¹·n≤ _qⁿ_k. Stated less technical, the proof of Lemma 7 is given by the fact that the hyperplane with the minimum number of k-subspaces contains at most as manyk-subspaces as the average number ofk-subspaces per hyperplane.

Taking also the third equation of (1) into account implies a quadratic criterion:

Lemma 8. Letm∈ZandNbe aq^r-divisible set ofk-subspaces inF^vq. Then,τ(n, q^r, q^k, m)·q^v−2k−2r−m(m−1)≥ 0, whereτ(n,∆, u, m) := ∆²u²m(m−1)−n(2m−1)u(u−1)∆ +n(u−1)(n(u−1) + 1).

PROOF. Withy =q^v−2k,u=q^k, and∆ =q^r, we can rewrite the equations of (1) tou²y−1 = (q−1)P

i∈Za_i, n·(uy−1) = (q−1)P

i∈Zia_i, andn(n−1)·(y−1) =P

i∈Zi(i−1)a_i.(n−m∆)(n−(m−1)∆)times the first minus2n−(2m−1)∆−1times the second plus the third equation givesy·τ(n,∆, u, m)−∆²m(m−1) = (q−1)P

i∈Z(n−m∆−i)(n−(m−1)∆−i)ai= (q−1)P

h∈Z∆²(m−h)(m−h+ 1)a_n−h∆≥0.

As a preparation we present another classification result:

Lemma 9. IfN is aq-divisible set ofk-subspaces inF^vq of cardinalityq^k+ 1, thenN partitionsF^2kq . PROOF. Settingci := (q−1)a1+iqandl :=q^k−1−1we can rewrite the equations of (1) toPl

i=0ci = q^v−1, Pl

i=0(1 +iq)ci = (q^k+ 1) q^v−k−1

, andPl

i=0iq(1 +iq)ci = (q^k+ 1)q^k q^v−2k−1

. Sinceql+ 1times the second minusql+ 1times the first minus the third equation gives0≤Pl

i=0iq²(l−i)ci=−q^k+1 q^v−2k−1 , we havev= 2k. Every point ofF^vq is covered by an element fromN due to2k

1

q/k 1

q =q^k+ 1.

3. PROOF OFHEDEN’S RESULTS AND FURTHER IMPROVEMENTS

LetPbe a vector space partition of typedlu_l

. . . d2u₂

d1u₁

ofF^v

0

q , whereu1, u2>0,dl>· · ·> d2> d1≥1. LetN be the set ofd1-subspaces andV be the subspace spanned byN. Bynwe denote the cardinality ofN and byaiwe denote the number of hyperplanes ofV that contain exactlyielements fromN.

Assume thatq^d2−d1does not divideu1. We have#(N ∩H)≥1for every hyperplaneH ofV due to Lemma 3, so that Lemma 7 givesu1≥q^d1. Thus, we haveu1 ≥q^d1+ 1. Ifu=q^d1+ 1then we can apply Lemma 9 for the classification of the possible setsN. Ifu₁<2q^d2−d1then fora_i>0we havei < q^d2−d1 andi≡u₁ (mod q^d2−d1) so that we can apply Lemma 2. Thus, eitherd₂dividesd₁andu₁= (q^d2 −1)/(q^d1−1)oru₁>2q^d2−d1. The first case can be attained by ad₂-spread where one d₂-subspace is replaced by ad₁-spread, see Example 5. We remark that no assumption on the relation betweend₂andd₁is used in our derivation. However, ifd₂ <2d₁thend₁cannot divided₂andq₁^d+ 1>2q^d2−d1.

Assume thatq^d2−d1dividesu1. Setting∆ =q^d2−d1,u=q^d1,n= ∆l, andm=l^†for some integerl, we conclude τ(n,∆, u, m) = ∆l(∆l−∆u+u−1)≥0from Lemma 8, so thatl≥

u−_∆^u +_∆¹

. The right-hand side is equal tou=q^d1 ifd2 ≥2d1and tou−u/∆ + 1 = q^d1−q^2d1−d2+ 1otherwise, which is equivalent ton≥ q^d2 and n≥q^d2−q^d1+q^d2−d1. We remark that equality is achievable in the latter case via the2-weight codes constructed in [1] (with parametersn⁰ =d1andm=d2−d1). We do not know whether the correspondingq^d2−d1-divisible set of d1-subspaces can be realized as a vector space partition ofF^vq.^‡ For the first case see Example 6.

†The choice formcan be obtained by minimizingτ(n,∆, u, m), i.e., solving∂τ(n,∆,u,m)

∂m = 0and rounding.

‡A suitable test case might be to decide whether a vector space partition of type4⁴3¹³⁵2⁶exists inF¹⁰2 .

The above comprises [4, Theorems 1-4]. Given the stated examples, just Theorem 1(ii), for the case whered1

does not divided2, leaves some space for improving the lower bound onu1. To that end we analyze Lemma 8 in more detail. Since the statements look rather technical and complicated we first give a justification for the necessity of this fact. Via the quadratic inequality of Lemma 8 intervals of cardinalities can be excluded for different values of the parameterm. However, some cardinalities are indeed feasible. Ifr = ak+bwith0 ≤b < kthen the two constructions from Example 5 and Example 6 giveq^r-divisible set ofk-subspaces of cardinality(a+2)k

1

q/k 1

q and q^k+r, respectively. Forq = 2,r = 3,k = 2the cardinalities of these two examples are given by21and32. In general, each twoq^r-divisible setsN1andN2ofk-subspaces can be combined to aq^r-divisible set ofk-subspaces of cardinality#N1+ #N2. Since(a+2)k

1

q/k 1

qandq^k+rare coprime there exists some integerFq(k, r)such that q^r-divisible sets ofk-subspaces exist for every cardinalityn > Fq(k, r). Below that number some cardinalities can be excluded, but their density decreases with increasingn. Our numerical example is continued after the proof of Theorem 12.

Proposition 10. LetN be aq^r-divisible set ofk-subspaces inF^vq,u=q^kand∆ =q^r. Then,n /∈h

1,^qk+r_qr−1⁻¹

and

n /∈ 1

u−1 ·

∆um−∆u+ 1 2 −1

2

√ω

, 1

u−1 ·

∆um−∆u+ 1 2 +1

2

√ω

, whereω= (∆u−2m)²+ 2∆u+ 1−4m²

, for allm∈Nwith2≤m≤_∆u

4 +¹₂+_4∆u¹ .

PROOF. We set∆ = ∆uandn=n(u−1)so thatτ(n,∆, u, m) = ∆²m(m−1)−n∆(2m−1) +n(n+ 1). We haveτ(n,∆, u, m) ≤0iff

n−∆m+^∆+1₂ ≤ ¹₂

q

∆²−4m∆ + 2∆ + 1andm ≤ ^∆₄ +¹₂ + ¹

4∆. Rewriting and applying Lemma 8 with1≤m≤_∆u

4 +¹₂+_4∆u¹

gives the result sincem(m−1)>0form≥2.

Proposition 11. LetN be aq^r-divisible set ofk-subspaces inF^vq, wherer=ak+bwitha, b∈N,0 < b < kand a≥1. Then,n≥ ^q(a+2)k_qk−1⁻¹ =q^r·q^k−b+^qr·q_qk^k−b−1⁻¹ = ∆q^k−b+q^kΘ + 1, where∆ :=q^randΘ := ^q_q^akk−1⁻¹. PROOF. From Lemma 2 we concluden≥2q^rand setu=q^k. For2 ≤m≤q^k−bwe have2∆u+ 1−4m² >0, so that Proposition 10 givesn /∈hl∆u(m−1)−1/2+m

u−1

m

,j_{∆um−1/2−m}

u−1

ki

. Since∆(m−1)≤l∆u(m−1)−1/2+m u−1

m

=

∆(m−1) +l

∆(m−1)−1/2+m u−1

m≤∆mand

j∆um−1/2−m u−1

k

= ∆m+mq^bΘ +j_mqb−1/2−m q^k−1

k

= ∆m+mq^bΘ, we concluden /∈

∆m,∆m+mq^bΘ

for2≤m≤q^k−b. It remains to shown /∈

∆m,∆m+mq^bΘ + 1,∆(m+ 1)−1

=:Imfor all2 ≤m ≤q^k−b−1. Ifn∈ Im, then we can writen= ∆m+mq^bΘ +xwithx≥1andmq^bΘ +x <∆, so thatq^k· mq^bΘ +x

= ∆m+mq^bΘ + xq^k−mq^b

<∆m+mq^bΘ +x=n, which contradicts Lemma 7.

In other words, in the case of Theorem 1(ii), whered2 = ad1+b with0 < b < d1 anda, b ∈ N, we have u1 ≥ q^d2−d1 ·q^d1−b+ ^q(a+1)d_qd1−1¹⁻¹ = ^q(a+2)d_qd1−1¹⁻¹, which can be attained by and1-spread inF^(a+2)dq ¹. Without the knowledge ofb, we can stateu₁≥q·q^d2−d1+l_qd2 +1−1

q^d1−1

m

, which also improves Theorem 1(ii) and is tight whenever d2+ 1is divisible byd1. Summarizing our findings we obtain our main theorem:

Theorem 12. For a non-emptyq^r-divisible setN ofk-subspaces inF^vq the following bounds onn= #N are tight.

(i) We haven≥q^k+ 1and ifr≥kthen eitherkdividesrandn≥ ^qk+r_qk−1⁻¹ orn≥ ^q(a+2)k_qk−1⁻¹, wherer=ak+b with0< b < kanda, b∈N.

(ii) Letq^rdividen. Ifr < kthenn≥q^k+r−q^k+q^randn≥q^k+rotherwise.

For (i) the lower bounds are attained byk-spreads, see Example 5. For (ii) the second lower bound is attained by a construction based on lifted MRD codes, see Example 6. In the other case the2-weight codes constructed in [1] attain the lower bound. Thus, Theorem 12 is tight and implies an improvement of Theorem 1(ii).

While the smallest cardinality of a non-emptyq^r-divisible set ofk-subspaces overFq has been determined, the spectrum of possible cardinalities remains widely unknown. Fork= 1[7, Theorem 12] states that eithern > rq^r+1 or there exist integersa, bwithn =ar+1

1

q +bq^r+1and bounds for the maximum excluded cardinality have been determined in [5]. However, Lemma 7 and Lemma 8, applied via Proposition 10, give restrictions going far beyond Theorem 12. For q = 2, r = 3, k = 2, andn ≤ 81 we exemplarily state that only n ∈ {21,31,32,33,42, 43,44,52, . . . ,55,62, . . . ,66,72, . . . ,78} might be attainable. The mentioned constructions cover the cases n ∈ {21,32,42,53,63,64,74} ⊆ {21a+ 32b : a, b ∈ N}. Replacing the lines by their contained3points, we obtain 2⁴-divisible sets of1-subspaces inF^vq of cardinality3n, for which two further exclusion criteria have been presented in [7], excluding the casesn∈ {33,44}. [7, Lemma 23] is based on a cubic polynomial obtained from (1) and (2), similar to the quadratic polynomial from Lemma 8 obtained from (1). Here, the presence ofkadditionalb_i-variables

may make the analysis more difficult fork > 1. For aq^r-divisible setN of 1-subspaces we have thatN ∩H is q^r−1-divisible for every hyperplaneH, which allows a recursive application of the linear programming method. For k >1we need to considerk-subspaces andk−1-subspaces inH, see [7, Section 6.3], which makes the bookkeeping more complicated.

The determination of the possible spectrum of cardinalities ofq^r-divisible sets ofk-subspaces remains an interesting open problem. Even for small parameters this might be challenging. A possible intermediate step is the determination of the number F_q(k, r) being similar to the Frobenius number. Extending the small list of constructions is also worthwhile.

ACKNOWLEDGEMENT

I am very thankful for the comments of two anonymous reviewers, which helped to improve the paper.

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DEPARTMENT OFMATHEMATICS, UNIVERSITY OFBAYREUTH, 95440 BAYREUTH, GERMANY E-mail address:sascha.kurz@uni-bayreuth.de