Remarks on Constant-Dimension Subspace Codes - ALCOMA 15

Remarks onConstant-DimensionSubspace

Codes

ThomasHonold

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GeneralConstructions

A StrangeInvariant forSubspaces

q-Ary(7; M; 4; 3)

subspacecodes

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Remarks on Constant-DimensionSubspace Codes

Thomas Honold

Department of Information Science and Electronics EngineeringZhejiang University

ALCOMA15Kloster Banz

March 16, 2015


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Outline

1 Network Coding

2 Subspace Codes

3 General Constructions

4 A Strange Invariant for Subspaces

5 q-Ary (7; M; 4; 3) subspace codes

6 References


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In Memoriam Axel Kohnert


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Outline

1 Network Coding

2 Subspace Codes



5 q-Ary (7;M; 4; 3) subspace codes

6 References


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The Basic Idea of Network CodingCoding replaces Routing

Instead of simply replicating some of the incoming informationpackets at an outgoing link (routing), network nodes compute theoutgoing packets as a function of the incoming packets (networkcoding).


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Optimal Routing Solution

t1 t2

1 2 3

s

2 1 3

4

3

1 1

2

3

t1 t2

1 2 3

s

b4b5 b3b1b2b3

b3b4b5

b3

b1b2b3

b4b5b1b2


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Optimal Coding SolutionCannot be achieved by routing

t1 t2

1 2

3

4

s

1 1

1

1

1

11

1 1

t1 t2

1 2

3

4

s

b1 b2

b1

b1

b2

b2b1 + b2

b1 + b2 b1 + b2


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Two-Way Wireless RelayChannel

Suppose two mobile stations want to send each other one bit ofinformation.

A BBase stationb1 b2

b1 + b2

Store and Forward� A sends b1 to the relay� B sends b2 to the relay� The relay broadcasts b1 (to A)� The relay broadcasts b2 (to B)

Requires 4 timeslots

Network Coding� A sends b1 to the relay� B sends b2 to the relay� The relay broadcasts b1 + b2

(to both A and B)

Requires only 3 timeslots


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The Main Results

Ahlwede-Cai-Li-Yeung (2000)Information can be transmitted from a single source node tomultiple sink nodes simultaneously at a rate equal to the minimumof the different unicast capacities (given by the Max-Flow Min-CutTheorem from Graph Theory).

This maximum transmission rate is called multicast capacity ofthe network.

Li-Yeung-Cai (2003)Linear Network Coding over a sufficiently large finite fieldachieves multicast capacity.

Ho-Médard-Koetter-Karger-Effros-Shi-Leong (2006)Random Linear Network Coding achieves multicast capacity withhigh probability.

Yeung-Li-Cai-Zhang (2006)Error-Correcting Network Codes in the “coherent” setting


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Linear Operator ChannelsKoetter-Kschischang (2008)

Consider a packet network capable of transmitting packets oflength v over a finite field Fq (i.e. elements of Fv

q ) along its edges(links) and employing (random) linear network coding.

X matrix having as its rows the packets injected by the source sinto the network.

Y matrix of packets (again considered as row vectors) receivedby a particular sink node t .

G Global Transfer Matrix of t (the matrix describing the overalllinear relation between source packets and packets receivedby t in the error-free case).

Z matrix of error packets added to the information packets at thevarious edges (one error packet for each edge of the network)

H Error Transfer Matrix of t

Random Linear Matrix ChannelY = GX + HZ with p(YjX) =

P

GX+HZ=Y p(G;H;Z) for someprobability distribution p(G;H;Z).


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Observation of Koetter-KschischangIn the error-free case (Z = 0) the row space hYi is a subspace ofhXi, and is equal to hXi if G has full column rank.

Moreover, under reasonable assumptions on the distribution ofthe encoding vectors at the network nodes and the error vectorsat the network links, the channel reduces to a subspace channel

p(V jU) =

X

Y=V

p(YjX); where X satisfies hXi = U:

EncodingThe source node selects a matrix X 2 F

k�vq , k = dim(U), such

that hXi = U and injects the rows of X as packets into the network.

DecodingA sink node tries to reconstruct the sent subspace U from thereceived subspace V = hYi, which is generated by the packetsreceived at the sink node (the rows of Y).


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Outline

1 Network Coding

2 Subspace Codes




6 References


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F

vq Vector space of v -tuples x = (x1; : : : ; xv), xi 2 Fq , over

the finite field Fq = GF(q) (“ambient space”, “packetspace”)

S = S(F

vq ) Lattice of subspaces of Fv

q

Sk Subset (“Grassmannian”) consisting of allk -dimensional subspaces in S (where 0 � k � v )

For U;V 2 S one defines

Subspace Distance

ds(U;V ) = dim(U + V )� dim(U \ V )

= dim(U) + dim(V ) � 2 dim(U \ V )

Injection Distance

di(U;V ) = maxfdim(U); dim(V )g � dim(U \ V )

q-ary (v ; M; d) Subspace CodeA subset C � S of size (cardinality) #C = M and minimumdistance (say, in the subspace metric)ds(C) := minfds(U;V );U;V 2 C;U 6= Vg = d


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Example setting (q = 2, v = 3)F

32 = f000; 100; 010; 001; 110; 101; 011; 111g.

F

32

h100i? h010i? h001i? h110i? h101i? h011i? h111i?

h100i h010i h001i h110i h101i h011i h111i

f000g

Hasse diagram of S(F32 )


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The Main Problem of Subspace CodingAkin to the Main Problem of Algebraic Coding Theory

The main problem of classical Algebraic Coding Theory asks forthe determination of the maximum dimension k of a linear [n; k ; d ℄code over Fq (i.e. a k -dimensional subspace of Fn

q with minimumHamming distance d).

Main Problem of Subspace CodingFor a given prime power q > 1 and given positive integers2 � d � v � 1 determine the maximum size M of a q-ary (v ;M; d)subspace code.

Constant-Dimension CodesA constant-dimension code is a subspace code all of whosemembers have the same dimension k ; notation (v ;M; d ; k).

For U;V 2 Sk we have ds(U;V ) = 2di(U;V ) = 2k � 2 dim(U \ V ).

Main Problem for Constant-Dimension CodesGiven q and 1 � Æ � k � v � 1, determine the maximum size Mof a q-ary (v ;M; 2Æ; k) constant-dimension code in (F

vq ; ds).


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TablesAq(v ; d) Maximum size of a q-ary (v ;M; d) subspace code

Aq(v ; d ; k) Maximum size of a q-ary (v ;M; d ; k)constant-dimension subspace code

In the constant-dimension case it suffices to consider (v ; d ; k)with 1 � k � v=2 (since C 7! C

?

= fU?

;U 2 Cg defines anautomorphism of the metric space (F

vq ; ds)) and d 2 f2; 4; : : : ; 2kg.

knd 2 4 6

1 632 651 213 1395 77 9

A2(6; d ; k)

knd 2 4 6

1 1272 2667 413 11811 329–381 17

A2(7; d ; k)

knd 2 4 6 8

1 2552 10795 853 97155 1326–1493 344 200787 4797–6477 257–289 17

A2(8; d ; k)


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Geometric View of Subspace Codes

A set C of k -dimensional subspaces of Fvq forms a (v ;M; 2Æ; k)

subspace code if and only if t = k � Æ + 1 is the smallest positiveinteger such that any t-dimensional subspace of Fv

q is containedin at most one member of C.

=) Constant-dimension codes may as well be defined by thisgeometric “packing condition”. The nontrivial range for t is0 � t � k � 2.

A standard double-counting argument gives

#C �

�

kt

�

q�

�

vt

�

q; or #C �

t�1Y

i=0

qv�i� 1

qk�i� 1

;

with equality if and only if each t-dimensional subspace of Fvq is

contained in precisely one codeword of C. Such structures areknown as Steiner systems over finite fields (q-analogues ofordinary Steiner systems) and form the “best” subspace codes(provided, of course, that they exist).


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Better Upper Bounds

Integrality ConditionsA Steiner system Sq(t ; k ; v) over Fq can only exist if the numbers

�s =

Qt�1i=s

qv�i�1

qk�i�1 are integers for 1 � s � t . This gives

#C �

�

qv� 1

qk� 1

�

qv�1� 1

qk�1� 1

: : :

�

qv�t+1� 1

qk�t+1� 1

�

: : :

��

:

Using available information on smaller casesThe recursive version of the above bound is

Aq(v ; d ; k) ��

qv� 1

qk� 1

� Aq(v � 1; d ; k � 1)�

:

This gives a better bound if a stronger bound forAq(v � 1; d ; k � 1) (or Aq(v � 2; d ; k � 2), Aq(v � 3; d ; k � 3),etc.) is known.


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Example (k = 3, t = 2)For a q-ary (v ;M; 4; 3) constant-dimension code the best knowngeneral upper bound is

#C �

8

<

:

j

(qv�1)(qv�1

�1)(q3

�1)(q2�1)

k

if v � 1 (mod 2);j

qv�1

q3�1

�

qv�1�q

q2�1 � q + 1

�k

if v � 0 (mod 2):

The sharper bound for v � 0 (mod 2) follows from the fact that a(v � 1;M; 4; 2) code (partial line spread in PG(v � 2; q)) has sizeat most

qv�3+ qv�5

+ � � �+ q3+ 1 =

qv�1� q

q2� 1

� q + 1:

Further the bound excludes the existence of q-analogues STSq(v)of Steiner triple systems in the case v � 5 (mod 6).

=) The necessary condition for the existence of an STSq(v) isv � 1; 3 (mod 6) (the same as for an ordinary STS(v)).


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Steiner Systems over Finite Fields

Known ResultsIn fact only very little is known about the existence problem forSq(t ; k ; v).

� The cases STSq(v), v 2 f7; 9g, any q, are all unsolved

� There exists an STS2(13) (in fact many non-isomorphicones) [2].

The known STS2(13) are invariant under the normalizer of aSinger group of PG(6; q), which has order(213

� 1)� 13 = 106 483. This greatly facilitates a computersearch and was the route to success.

Singer-invariant STSq(7) do not exist for the first few small valuesof q. This perhaps explains why the case v = 13 was solved first(for q = 2).


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Outline

1 Network Coding

2 Subspace Codes




6 References


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The Lifting ConstructionSilva-Kschischang-Koetter (2008)

The lifting construction yields q-ary�

v ; q(v�k)(k�Æ+1); 2Æ; k

�

constant-dimension codes of fairly large (though not optimal) size.

IdeaRestrict attention to the k -dimensional subspaces U of Fv

q whosecanonical matrices have their pivots in the first k columns; i.e.U = h(Ik jA)i for some A 2 F

k�(v�k)q . For such subspaces we have

ds(U;V ) = 2 dim(U + V )� 2k

= 2 rk�

Ik AIk B

�

� 2k = 2 rk�

Ik A0 B� A

�

� 2k

= 2 rk(B� A) = 2dr(A;B);

where dr denotes the so-called rank distance.

=) A matrix code A = fA1; : : : ;AMg � F

k�(v�k)q of minimum rank

distance dr(A) = Æ yields a (v ;M; 2Æ; k) constant-dimension codeC = fU1; : : : ;UMg via Ui = h(Ik jAi)i.


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MRD CodesThe maximum size of a matrix code A � F

k�lq with minimum rank

distance Æ 2 f1; 2; : : : ;min(k ; l)g is fortunately known. In order tosimplify the statement of the result, we assume k � l. (This is norestriction, since we can transpose everything.)

Theorem (Delsarte-Gabidulin-Roth, 1978–1991)The maximum size of a q-ary (k ; l; Æ) rank-distance code isq(k�Æ+1)l . A particular example of such a code is the Gabidulincode

G = fx 7! a0x + a1xq+ a2xq2

+ � � �+ ak�Æxqk�Æ

; ai 2 Fql g;

defined in a basis-independent manner as a space of Fq -linearmaps W ! Fq l on a k-dimensional Fq -subspace W of Fql .

Notes� Matrix codes meeting the bound in the Theorem are called

maximum rank distance (MRD) codes. MRD codes areanalogous to maximum distance separable (MDS) codes inHamming spaces, and the Gabidulin codes can be seen asq-analogs of Reed-Solomon codes.


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Notes cont’d� Polynomials in Fq l [X ℄ of the special form

a(X ) = a0X + a1Xq+ a2Xq2

+ � � �+ anXqnare called

linearized polynomials or q-polynomials. They defineFq -linear endomorphisms of Fq l [X ℄ via x 7! a(x), since(x + y)q

= xq+ yq for x ; y 2 Fql and xq

= x for x 2 Fq .

� dr(G) � Æ follows from the degree bound for polynomials: Anonzero map x 7! a0x + a1xq

+ � � �+ ak�Æxqk�Æ

has at mostqk�Æ zeros in Fq l , i.e. a kernel of dimension � k � Æ, andhence rank at least Æ.

�

#A = q(k�Æ+1)l is reflected in the following characteristicproperty of q-ary (k ; l; Æ) MRD codes: With r := k � Æ + 1 wehave that for any full-rank matrix Z 2 F

r�kq the map A ! F

r�lq ,

A ! ZA is a bijection.


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Example (q = 2, k = l = 3)The maximum size of a set of binary 3� 3 matrices of rankdistance 3 (equivalently the sum of any two distinct matrices isnon-singular) is 21�3

= 8. As a particular example we can take amaximal subfield (�

=

F8 ) of the matrix ring M3(F2 ), for example

G1 =

8

<

:

0

�

a c b + cb a cc b + c a + b + c

1

A

; a; b; c 2 F2

9

=

;

;

the subfield generated by the companion matrix A of X3+ X + 1.

If we stipulate instead rank distance 2, the maximum size is22�3

= 64, realized by

G2 = fa0x + a1x2; a0; a1 2 F8g � End(F8=F2 ):

Using the same basis of F8 as above and denoting the matrix ofthe Frobenius automorphism F8 ! F8 , x 7! x2 by F, we haveG2 � G1, and G2 is generated over F2 by the 6 matrices I, A, A2, F,AF, AF2.


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The Echelon-FERRERS ConstructionFrom now on we will focus on the determination of the best (i.e.largest) (7;M; 4; 3) constant-dimension codes and use the caseq = 2 as a showcase.

Applying the lifting construction to G2 (Æ = 2 =) d = 4), we obtaina binary (7; 256; 4; 3) constant-dimension code C. The 256canonical matrices representing the subspaces in C have shape

0

�

1 0 0 � � � �

0 1 0 � � � �

0 0 1 � � � �

1

A

:

In projective geometry terms, C consists of 256 planes in PG(6; 2)mutually intersecting in at most a point (since distinct subspacesin C intersect at most in a 1-dimensional subspace of F7

2 ).Moreover, all planes in C are disjoint from the special solidS = (000 � � � �).

QuestionHow far can C be extended?The answer (M = 291) is provided by an extension of theechelon-Ferrers construction and a “geometric” upper bound.


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The Upper Bound

LemmaThe planes in C cover every line disjoint from S exactly once.

In particular, the number of lines in PG(6; 2) disjoint from S mustbe 7� 256 = 1792 (since every plane containes 7 lines). This canbe checked independently by a counting argument.

Proof.A line L disjoint from S has a canonical matrix of the form (ZjB)

with Z 2 F

2�32 , B 2 F

2�42 and rk(Z) = 2 (in fact Z must be itself

canonical).By what we have seen, there exists A 2 G2 such that ZA = B.=) Z(I3jA) = (ZjZA) = (ZjB), implying that L is contained in theplane of C with canonical matrix (I3jA).

From the lemma the upper bound M � 291 is now easy to derive:We cannot add a plane to C meeting S in a point (since such aplane would contain lines disjoint from S and lead to a multipleline cover. Hence the best we can do is adding

�42

�

2 = 35 planesto C, meeting S in the 35 lines contained in S.


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The ConstructionFirst we describe the general idea of the echelon-Ferrersconstruction in our special setting q = 2, (v ; k ; d) = (7; 3; 4).

Identifying vectorTo a 3-dimensional subspace U of F7

2 assign the vector v(U) 2 F

72

having 1’s in the positions of the pivot columns of U and 0’selsewhere. The vector v(U) has Hamming weight 3 anddetermines the shape of the canonical matrix of U. It is calledidentifying vector of U.

Key ObservationThe subspace distance ds(U;V ) is lower-bounded by theHamming distance dHam

�

v(U); v(V )

�

.

Echelon-Ferrers construction (Etzion-Silberstein 2009)

1 Select a set I of identifying vectors of constant Hammingweight 3 and minimum Hamming distance 4.

2 For each v 2 I determine the largest (3; 4; 2) rank distancecode with the particular shape imposed by v and lift it to asubspace code C(v). The union

S

v2I C(v) then has d = 4.


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Specifically we chooseI = f1110000; 1001100; 1000011; 0101010; 0100101; 0011001; 0010110g,corresponding to the 7 shapes0

�

1 0 0 � � � �0 1 0 � � � �0 0 1 � � � �

1

A

;

0

�

1 � � 0 0 � �0 0 0 1 0 � �0 0 0 0 1 � �

1

A

;

0

�

1 � � � � 0 00 0 0 0 0 1 00 0 0 0 0 0 1

1

A

;

0

�

0 1 � 0 � 0 �0 0 0 1 � 0 �0 0 0 0 0 1 �

1

A

;

:

0

�

0 1 � � 0 � 00 0 0 0 1 � 00 0 0 0 0 0 1

1

A

;

0

�

0 0 1 0 � � 00 0 0 1 � � 00 0 0 0 0 0 1

1

A

;

0

�

0 0 1 � 0 0 �0 0 0 0 1 0 �0 0 0 0 0 1 �

1

A

:

From the first shape we can select 28= 256 matrices (as we have

seen), from the second shape 24= 16 matrices (project the first

256 onto the lower left 2� 2 square and use the kernel), etc. Weobtain a binary (7; 289; 4; 3) constant dimension code, asasserted.

v 1110000 1001100 1000011 0101010 0100101 0011001 0010110P

#C(v) 28 24 20 23 21 22 21 289


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How to Achieve M = 291 ?

This can be done using the concept of pending dots [1], arefinement of the echelon-Ferrers construction.

Alternative ConstructionA plane E meeting S in a line L has the form E = hP; Li = P + Lfor some point P outside S; F = hE ;Si is a 4-flat above S.

#4-flats above S = 7

#lines in S = 35 = 7� 5

=) It is reasonable to choose 5 points in every F and connectthem to 5 lines in S.

Condition: E1 \ E2 \ (P n S) 6= ; =) L1 \ L2 = ;

Now recall that PG(S)

�

=

PG(3; 2) admits a decomposition of its35 lines into 7 spreads (called a packing or parallelism).

=) We can match the 4-flats above S to the spreads and connecta fixed point in F n S to the 5 lines of the corresponding spread.


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Outline

1 Network Coding

2 Subspace Codes




6 References


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Definition of the Æ-invariantRepresent PG(3; q) = PG(F

4q =Fq ) as PG(Fq4 =Fq ).

DefinitionFor a; b 2 Fq4 set Æ(a; b) = abq

� aqb = aQ

�2Fq(b� �a) and for a

line L = ha; bi of PG(Fq4 =Fq )

Æ(L) = Fq Æ(a; b):

Notes

�

Æ(L) is simply the product of all points on L in Fq4 =F�

q . HenceL 7! Æ(L) maps lines to points.

�

Æ(L) is the projective version of the 2nd Dickson line invariantfrom Modular Invariant Theory.

�

Æ(E) for a plane E can be defined as in Note 1 (but not as inthe definition).


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Properties of Æ

� b 7! Æ(a; b) is Fq -linear with kernel Fq a, hence maps linepencils (the set of lines through a point Fq a) bijectively ontothe points of a plane (the plane with equationT(a�q�1x) = 0).=) Æ is one-to-one on the lines in every plane.

�

Æ maps the lines in a plane bijectively to the points of anotherplane.


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Outline

1 Network Coding

2 Subspace Codes




6 References


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Previous ConstructionsM Construction Ref.q8 lifting construction [3]

q8+ q4

+ q3+ q2

+ 2q + 1 echelon-Ferrers [1]q8

+ q4+ q3

+ 2q2+ q + 1 pending dots [1]

LMRD BoundM � q8

+

�42

�

q= q8

+ q4+ q3

+ 2q2+ q + 1

New Result (H.-Kiermaier, 2015)

1 There exists a q-ary (7; q8+ q5

+ q4� q � 1; 4; 3) subspace

code C with the following additional properties:

(i) The planes in C meet a fixed 3-flat S of PG(6; q) in atmost a point.

(ii) C is invariant under a Singer group of S (acting triviallyon a complementary plane).

2 There exists a ternary (7; 6976; 4; 3) subspace code.


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Notes on the new result

� 38+ 35

+ 34� 3� 1 = 6881

In Part (2) it is claimed that in the ternary case the code C ofPart (1) can be augmented by 34

+ 32+ 3 = 95 further

planes meeting S in a line.

� 28+ 25

+ 24� 2� 1 = 301. In the binary case the code C of

Part (1) can be augmented by 28 further planes meeting S ina line to a (7; 329; 4; 3) code [1], the largest size currentlyknown [3].

� The code C of Part (1) contains a distinguished subcode C0

with #C0 = q8+ q5

� q = #C � (q4� 1), which can be

augmented by q4� 1 further planes in myriads of ways.

Thus C is far from being the unique code of its size.

� The proof uses the same approach as in the firstcomputer-free construction of an optimal (6; 77; 4; 3)subspace code in [1] but introduces several new concepts.


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Putative q-Analogues of the Fano PlaneForming the motivation for this work

For a q-ary (7;M; 4; 3) subspace code C we have the bound

#C � q8+ q6

+ q5+ q4

+ q3+ q2

+ 1

with equality iff C is an STSq(7) (i.e. every line of PG(6; q) iscontained in a unique plane of C).

Structure of a putative STSq(7)

1 #C = q8+ q6

+ q5+ q4

+ q3+ q2

+ 1, with q4+ q2

+ 1planes in C passing through any point of PG(6; q).

2 The intersection vector �(S) = (�0; �1; �2; �3),�i = #fE 2 C; dim(E \ S) = ig takes the 2 values

(q8� q7

+ q3; q7

+ q6+ q5

� q3� q2

� q; q4+ q3

+ 2q2+ q + 1; 0);

(q8� q7

; q7+ q6

+ q5; q4

+ q3+ q2

; 1)

In particular we can arrange things such that S contains noplane of C and q8

� q7+ q3 planes disjoint from S.


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Overview of the Proof

Starting configurationA lifted (7; q8

; 4; 3) LMRD code L.

� In matrix terms, L consists of q8 matrices A 2 F

3�4q at

pairwise rank distance � 2 (the corresponding MRD code),each bordered by the 3� 3 identity matrix to yield thecanonical matrix (I3jA).

� In geometric terms, L consists of q8 planes in PG(6; q) thatare disjoint from the special solid S = (0; 0; 0; �; �; �; �) (thesolid with equation x1 = x2 = x3 = 0) and cover each linedisjoint from S exactly once.

The LMRD code constraintThe lines disjoint from S are bound by L.Any plane E meeting S in a point contains such lines, hencecannot be added to L. This leads to the LMRD code bound

M � q8+#lines in S = q8

+

�42

�

q = q8+ q4

+ � � � .


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ConsequenceIn order to overcome the LMRD code bound, some planes from L

must be removed. This frees some lines disjoint from S, possiblymaking enough space for new planes that meet S in a point to beadded.

Why this might workRemoving M1 planes from L frees M1(q2

+ q + 1) lines.If the free lines can be exactly covered by “new” planes meeting Sin a point, the number of new planes will be M1(q2

+ q + 1)=q2.

=) The overall code size has increased byM1(q2

+ q + 1)=q2�M1 = M1(q�1

+ q�2) > 0.

Questions� Which subsets L1 � L have the property that the lines

covered by the planes in L1 can be exactly rearranged intoplanes meeting S in a point?

� Which of these rearrangements satisfy the subspacedistance condition (equivalently, cover no line through a pointof S twice)?

A necessary condition for an exact rearrangement is q2j M1.


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Condition for ExactRearrangement

The planes in L are transversal to the q2+ q + 1 hyperplanes H

above L.

Every hyperplane section L ! PG(H), E 7! E \ H can bedescribed, in matrix terms, as (IjA) 7! (ZjZA) with a canonical2� 3 matrix Z (Z = cm(Z ), where Z is the 2-dimensionalsubspace of F3

q determined by H = Z � S).

By the MRD code property, the section determines the set of alllines disjoint from S in PG(H).

Fact (from the Geometry of Matrices)The constant-rank 1 subspaces of Fm�n

q are of the following twotypes.

1 All matrices with a fixed 1-dimensional row space Fq u � F

nq .

2 All matrices with a fixed 1-dimensional column spaceFq v � F

mq .


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LemmaSuppose L arises from a linear MRD code A � F

3�4q and H is a

hyperplane of PG(6; q) containing S with cm(H) =

�

Z 00 I4

�

. Thenfor R � A with #R = q2 the following are equivalent:

1 The q2 planes in L corresponding to R determine a plane inH that meets S in a point P.

2 ZR = B0 + U for some constant-rank 1 subspace U � F

2�4q

of the first type.

3 R = A0 +D for some A0 2 A and some 2-dimensionalFq -subspace D of A with the following properties: D hasconstant rank 2, and the (1-dimensional) left kernels of thenonzero members of D generate the row space Z of Z.

If these conditions are satisfied then the new plane N hascm(N) =

�

Z ZA00 s

�

, where s 2 F

4q is a generator of the common

1-dimensional row space of the nonzero matrices in ZD.Moreover, N \ S = Fq s.

Important NoteIn view of the MRD code property of A, there exists precisely onesuch matrix space D = D(Z ;P) (for each pair Z and P).


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For an exact rearrangement of the planes determined by R, theconditions of the lemma must be satisfied simultaneously for allq2

+ q + 1 hyperplanes H above S (or the corresponding Z ).

TheoremLet R be a t-dimensional Fq -subspace of a q-ary linear (3; 4; 2)MRD code A, having the following properties:

1 For each 2-dimensional subspace Z of F3q there exists a

1-dimensional subspace P = Z 0 of F4q such that

D(Z ;P) � R.

2 The map Z 7! Z 0 is one-to-one.

3 rk�

ZA1�ZA2s

�

= 3 whenever A1, A2 are in different cosets ofD(Z ;P) in R, where P = Z 0, Z = cm(Z ), and s = cm(P).

Then the (q2+ q + 1)qt “free” lines contained in the planes

corresponding to R can be rearranged into (q2+ q + 1)qt�2 new

planes meeting S in a point and such that the set N of newplanes has minimum subspace distance 4. Consequently, theremaining q8

� qt planes in L and the new planes in N constitutea (7; q8

+ qt�1+ qt�2

; 4; 3)q subspace code.


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Questions

1 Are there such subspaces R at all?

2 Provided the answer to Question 1 is yes, what is themaximal dimension t of such a subspace (clearly, 3 � t � 5).

ObservationThe rearrangement condition (Condition (1) of the theorem) isinherited to unions of cosets of R in A.


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Answers

For the Gabidulin code the answers turns out to be “yes” andt = 4. Moreover, there are “canonical” 3-and 4-dimensionalsubspaces T , R with these properties, from which every furthersuch subspace is obtained by “rotation” with a Singer cycle of S.(The number of choices for T and R is therefore q4

�1q�1 .)

=) There exist q-ary (7; q8+ q3

+ q2; 4; 3) subspace codes

consisting of planes meeting S in at most a point.

These codes are extendable to(7; q8

+ q4+ 2q3

+ 3q2+ q + 1; 4; 3) subspace codes using the

packing construction described earlier.


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The Basis-Free View . . .and how the extension field Fq4 comes into play

IdeaTake the ambient space of PG(6; q) as W � Fq4 , where

W = fx 2 Fq4 ; x + xq+ xq2

+ xq3= 0g is the trace-zero subspace

of Fq4 =Fq .

Special solid S = f0g � Fq4 , Fq4

3� 4 matrices Hom(W ; F

4q ) = fx 7! a0x+a1xq

+a2xq2; ai 2 Fq4 g

Gabidulin code G = fa0x + a1xq; a0; a1 2 Fq4 g.

LMRD code L = f�f ; f 2 Gg, where �f =�

(x ; f (x)); x 2 W

denotes the graph of f (as in Real Analysis)

Arbitrary 3-dimensional subspaces of W � Fq4 are parametrizedas U =

�

(x ; f (x) + y); x 2 Z ; y 2 T ; f 2 Hom(Z ; Fq4 =T )

=

U(Z ;T ; f ), where

Z =

�

x 2 W ; 9y 2 Fq4 such that (x ; y) 2 U

� W ;

T =

�

y 2 Fq4 ; (0; y) 2 U

� Fq4 :


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The Basis-Free View cont’dIncidence U(Z 0

;T 0

; f 0) � U(Z ;T ; f ) if and only if Z 0

� Z , T 0

� Tans f jZ 0

� f 0 2 Hom(Z 0

;T )

Our special subspaces of G

G = fa0x + a1xq; a0; a1 2 Fq4 g;

R = faxq� aqx ; a 2 Fq4 g;

T = faxq� aqx ; a 2 Wg;

D(Z ;P) = s(abq� aqb)�1

haxq� aqx ; bxq

� bqxi

for a 2-dimensional subspace Z = ha; bi of W and a pointP = Fq (0; s) of the special solid S (i.e. s 2 F

�

q4 ).

Notes� axq

� aqx = aQ

�2Fq(x � �a) has kernel Fq a

�

D(Z ;P) = ff 2 G; f (Z ) � Fq sg, as follows by scaling fromD

�

Z ; Fq (abq� aqb)

�

= haxq� aqx ; bxq

� bqxi

=) R; T have the properties required in the theorem.


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Going Beyond the Theorem

First IdeaRemove disjoint unions of “rotated” cosetsr(f + T ) =

�

(r(axq� aqx ; a 2 W

from G, where r 2 F

�

q4 , f 2 G.

Computer experiments [1] for the case q = 2 showed that at most4 rotated cosets of T can be used. This gives a binary subspacecode of size M = 256 + 4 � 6 + 34 = 314.

Key ideaDrop the requirement of an exact rearrangement, but make theremoval step invariant under the collineation group � of PG(6; q)consisting of all transformations (x ; y) 7! (x ; ry), r 2 F

�

q4

(essentially a Singer subgroup of S).

Then it suffices to check how many new planes through oneparticular point of S, say P1 = Fq (0; 1), can be added inaccordance with Condition (3) of the theorem.

If N1 new planes through P1 can be added, then#N = N1(q3

+ q2+ q + 1) new planes can be added in total.


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Notes

� Initially this idea was used for q = 2, in which case there isonly one candidate for the removed set: All cosets r(R n T ),r 2 F

�

16 (since R n T is a single coset).

The no. of removed planes is 15 �#(T n R) = 15� 8 = 120(exactly what would be needed for a 2-analogue of the Fanoplane, coordinatized such that no plane contained in S).

The no. of new planes through P1 that can be added wasfound by computer to be 11 (out of 14 candidate planes),with 4 different choices for the 11-set.=) M = 256� 120 + 15� 11 + 28 = 329.

� For q > 2 we remove all q4� 1 = (q � 1)� q4

�1q�1 rotated

cosetsr(f + T ); f 2 R n T ; r 2 F

�

q4 =F�

q :

This is also the size of the corresponding plane subset of aputative q-analogue of the Fano plane.


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The Collision GraphDefinitionThe collision graph � has as its vertices the q4

� q “candidatenew planes” through P1 that can be added to the expurgatedLMRD code without introducing a multiple cover of a line disjointfrom S (the relaxed rearrangement condition).

Two vertices (new planes) are adjacent in � if they have a linethrough S (or a point outside S) in common.The maximal sets of new planes through P1 that can be added tothe expurgated LMRD code are precisely the maximalindependent sets (cocliques) of the collision graph.

Notes� A collision graph may be defined w.r.t. any �-invariant set of

removed planes (and also LMRD codes other than theGabidulin code, “non-standard” rearrangements of the freelines into new planes, etc.)

� In the case q = 2 considered initially the collision graphturned out to be a K4 (which has 4 maximum cocliques) byinspection.


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Analysis of the Collision GraphThe end of the story

Step 1Find the representation N = N(Z ;T ; g) of the vertices of �.The result is:

N = N(Z ;P1; g), where Z � W is 2-dimensional and g : Z ! Fq4

is of the form

g(x) = Æ(a; b)�1(�(dxq

� dqx) + �(cxq� cqx)) =

Æ(�d + �c; x)Æ(a; b)

with � 2 F

�

q , � 2 Fq (q4� q choices for N).

Step 2Translate the collision condition into Algebra.The points outside S covered by N are

Fq

�

x ;Æ(�d + �c; x)

Æ(a; b)+ �

�

; Fq x 2 Z ; � 2 Fq ;

alltogether (q + 1)q choices.


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New planes corresponding to the same point Fq (d + �c) (q � 1different choices for �) do not collide, and we can reduce theanalysis to collisions between two such plane “bundles”.

Geometric ViewFq (d + �c) and Z generate a plane E of PG(Fq4 =Fq )

�

=

PG(3; q),which meets the trace-zero plane W in a line. This sets up abijection between the q4

�qq�1 = q3

+ q2+ q plane bundles and the

planes E 6= W of PG(Fq4 =Fq )

Then use that x 7! (xq� x)q�1 forms a complete invariant for the

orbits of AGL(1; q) = fx 7! �x + �;� 2 F

�

q ; � 2 Fq g acting on Fq4 .

ResultTwo plane bundles N(Z ;P1; F

�

q g), N(Z 0

;P1; F�

q g0) represented byplanes E ;E 0 of PG(Fq4 =Fq ) have collisions between them if andonly if

Æ(E)

Æ(Z )

q+1 =

Æ(E 0

)

Æ(Z 0

)

q+1 :

=) The collision graph between planes bundles is a union ofcomplete graphs, one for each value taken by the �-invariant�(E) = Æ(E)=Æ(Z )

q+1.


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Step 3Explicit computation of the �-invariant.Use that every plane of PG(Fq4 =Fq ) has the form E = aW forsome a 2 F

�

q4 (e.g. by Singer’s Theorem) and Z = W \ aW . Thequantities Æ(W ) and Æ(Z ) = Æ(W \ aW ) can be determined fromthe corresponding subspace polynomials.

ResultFor a plane E = aW 6= W of PG(Fq4 =Fq ) we have

�(E)

q�1= 1�

a(q�1)(q2+1)

� 1aq�1

� 1:

This shows that there are always collisions, viz. between planesof the form E = aq+1W , for which �(E)

q�1= 1 and hence

�(E) = Fq .

There are q2+ 1 such planes (including W ), forming a dual ovoid

(more precisely, a dual elliptic quadric) in PG(Fq4 =Fq )�

=

PG(3; q).

=) No q-analogue of the Fano plane can be constructed in thisway.


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Last StepDetermine the values of E 7! �(E) and their multiplicities.

LemmaE 7! �(E) maps all dual ovoid planes to the point Fq = Fq 1(hence the multiplicity of Fq is q2) and is one-to-one on thecomplementary set of planes.

Hence the collision graph on bundles of new planes consists of aKq2 and q3

+ q isolated vertices and has maximum coclique sizeq3

+ q + 1.

Proof of the lemma.The statement is equivalent to

x � 1y � 1

6=

xq2+1� 1

yq2+1� 1

(?)

for any pair of distinct elements x ; y 2 F

�

q4 that are (q � 1)-th

powers but not (q2+ 1)-th roots of unity.

Assume by contradiction that equality holds in (?) for some pairx ; y .


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Proof cont’d.

=)

x � 1y � 1

=

�

x � 1y � 1

�q2

=

xq2� 1

yq2� 1

;

since right-hand side of (?) is in the subfield Fq2 .

The two equations can be rewritten as

q2X

i=0

x i=

xq2+1� 1

x � 1=

yq2+1� 1

y � 1=

q2X

i=0

y i;

q2�1X

i=0

x i=

xq2� 1

x � 1=

yq2� 1

y � 1=

q2�1X

i=0

y i;

and together imply xq2= yq2

and hence x = y ; contradiction.


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Theorem (H.-Kiermaier, 2015)

1 There exists a q-ary (7; q8+ q5

+ q4� q � 1; 4; 3) subspace

code C with the following additional properties:

(i) The planes in C meet a fixed 3-flat S of PG(6; q) in atmost a point.

(ii) C is invariant under a Singer group of S (acting triviallyon a complementary plane).

2 There exists a ternary (7; 6973; 4; 3) subspace code.

Proof.(1)–(3) It only remains to check the size of C.

#C = q8� (q4

� 1)q3+ (q3

+ q + 1)(q � 1)(q3+ q2

+ q + 1)

= q8+ (q4

� 1)(q + 1) = q8+ q5

+ q4� q � 1

(4) (Non-exhaustive) computer search for extensions of thedistinguished (7; 6801; 4; 3) subcode C0.


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Further Notes/Open Problems� It can be shown that C can be extended by q2

+ 1 furtherplanes meeting the special solid S , Fq4 in the lines of aspread (in fact the standard spread formed by the1-dimensional subspaces of Fq4 =Fq2 ).

� The extension problem of C0 is closely related to the maximalsize of partial spreads contained in the q different regularSinger line orbits of PG(3; q). What is known about thesesizes?

� So far the extension problem for C or C0 has been solved onlyin the smallest case q = 2, where the answer is M = 329.

� Generalize our approach to other subspace codeparameters. The case v = 8, k = 3 (and d = 4) seems themost appropriate. M.Sc. student J. Ai of Zhejiang Universityhas shown by experimental study that for q = 2 a subspacecode of size M = 1286 can be obtained in this way.

� Find q-analogues of the Fano plane by applying our methodto other MRD codes than the Gabidulin code, or toappropriate smaller-than-MRD codes.


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Thank You


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Outline

1 Network Coding

2 Subspace Codes




6 References


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References

R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung.

Network information flow.

IEEE Transactions on Information Theory, 46(4):1204–1216,July 2000.

M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy, andA. Wassermann.

Existence of q-analogs of Steiner systems.

Preprint arXiv:1304.1462 [math.CO], Apr. 2013.

M. Braun and J. Reichelt.

q-analogs of packing designs.

Journal of Combinatorial Designs, 22(7):306–321, July 2014.

Preprint arXiv:1212.4614 [math.CO].

P. Delsarte.

Bilinear forms over a finite field, with applications to codingtheory.

Journal of Combinatorial Theory, Series A, 25:226–241,1978.


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T. Etzion and N. Silberstein.

Error-correcting codes in projective spaces via rank-metriccodes and Ferrers diagrams.

IEEE Transactions on Information Theory, 55(7):2909–2919,July 2009.

E. M. Gabidulin.

Theory of codes with maximum rank distance.

Problems of Information Transmission, 21(1):1–12, 1985.

T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi,and B. Leong.

A random linear network coding approach to multicast.

IEEE Transactions on Information Theory,52(10):4413–4430, Oct. 2006.

T. Honold and M. Kiermaier.

On putative q-analogues of the Fano plane and relatedcombinatorial structures.

Preprint, Feb. 2015.


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T. Honold, M. Kiermaier, and S. Kurz.

Optimal binary subspace codes of length 6, constantdimension 3 and minimum subspace distance 4.

To appear in the proceedings volume of the 11th InternationalConference on Finite Fields and their Applications(Magdeburg, July 22–26, 2013), AMS ContemporaryMathematics Series, Vol. 632, 2015. PreprintarXiv:1311.0464 [math.CO], Nov. 2013.

R. Koetter and F. Kschischang.

Coding for errors and erasures in random network coding.

IEEE Transactions on Information Theory, 54(8):3579–3591,Aug. 2008.

S.-Y. R. Li, R. W. Yeung, and N. Cai.

Linear network coding.

IEEE Transactions on Information Theory, 49(2):371–381,Feb. 2003.


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H. Liu and T. Honold.Poster: A new approach to the main problem of subspacecoding.In 9th International Conference on Communications andNetworking in China (ChinaCom 2014, Maoming, China,Aug. 14–16), pages 676–677, 2014.Full paper available as arXiv:1408.1181 [math.CO].

R. M. Roth.Maximum-rank array codes and their application to crisscrosserror correction.IEEE Transactions on Information Theory, 37(2):328–336,Mar. 1991.Comments by Emst M. Gabidulin and Author’s Reply, ibid.38(3):1183, 1992.

D. Silva, F. Kschischang, and R. Koetter.A rank-metric approach to error control in random networkcoding.IEEE Transactions on Information Theory, 54(9):3951–3967,Sept. 2008.


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A.-L. Trautmann and J. Rosenthal.

New improvements on the Echelon-Ferrers construction.

In A. Edelmayer, editor, Proceedings of the 19th InternationalSymposium on Mathematical Theory of Networks andSystems (MTNS 2010), pages 405–408, Budapest, Hungary,5–9 July 2010.

Reprint arXiv:1110.2417 [cs.IT].

R. W. Yeung, S.-Y. R. Li, N. Cai, and Z. Zhang.

Network coding theory.

Foundations and Trends in Communications and InformationTheory, 2(4/5):241–381, 2006.

Remarks on Constant-Dimension Subspace Codes - ALCOMA 15

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