Subspace Codes for Random Networks Based on Plücker Coordinates and Schubert Cells

0.17 MB·English

by  Anirban Ghatak

Subspace Codes for Random Networks Based on
Plu¨cker coordinates and Schubert Cells
Anirban Ghatak
Department of ECE
Indian Institute of Science, Bangalore, India-560012
Email: [email protected]
Abstract—Theconstructionofcodesintheprojectivespacefor furtheremployedinconjunctionwithalexicographicordering
3 error control in random networks has been the focus of recent of subspaces ([4]). The above technique and its variants ([6])
1 research. The Plu¨cker coordinate description of subspaces has
have resulted in some of the best-known constant dimension
0 been discussed in the context of constant dimension codes, as
2 well astheSchubertcell descriptionof certain codeparameters. subspace codes till date. Other important constructions of
n In this paper we use this classical tool to reformulate some constant dimension subspace codes include those presented
a standard constructions of constant dimension codes and give a in [7], [8], [9].
J unifiedframework.Wepresentageneralmethodofconstructing Restriction of codeword subspaces to a particular Grassman-
non-constantdimensionsubspacecodeswithrespecttominimum
9 nian yields constant dimension subspace codes; a natural
subspace distance or minimum injection distance as the union
2 generalizationwastoattempttheconstructionofnon-constant
of constant dimension subspace codes restricted to selected

Schubert cells. The selection of these Schubert cells is based dimension subspace codes overthe entire projective space. In
]
T on the subset distance of tuples corresponding to the Plu¨cker this context, a new metric, namely the injection distance, was
I coordinate matrices associated with the subspaces contained in definedin[10],whichwasshowninsomecases(forinstance,
. the respective Schubert cells. In this context, we show that a
s aworst-caseadversarialmodel)togiveamoreprecisemeasure
c recent construction of non-constant dimension Ferrers-diagram of the performance of a non-constant dimension code than
[ rank-metric subspace codes is subsumed in our framework.
the subspace distance metric. In [11], subspace codes (non-

2 I. INTRODUCTION constant dimension) were constructed for the injection metric
v A linear multicast network coding model is defined as based on the methodsin [3]. Boundson the size of projective
2
follows: A source node transmits n packets, each an m-ary space codes, analogousto classical codingtheoreticones, and
6
symboloverF ; each nodein the networktransmits F -linear some explicit constructions of non-constant dimension codes
3 q q
6 combinationsoftheincomingpacketsatthatnode,andatany were presented in [12].
. destination node the received packets may be represented as Anewparadigminrandomnetworkcodingwasintroducedby
1
0 the rows of an N ×m matrix Y = AX +DZ, where the the so-called orbit codes [13], which generalizes some of the
3 matricesX,Z are ofrespectivedimensions n×m,t×m over resultsin[7]anddescribesconstantdimensionsubspacecodes
1 F . The matrix X has n source packets as its rows, A is the as orbits of the action of suitable subgroups of the general
q
:
v transfer matrix of the network, the rows of Z are the error lineargrouponaGrassmannian.Subsequentwork([14],[15],
i packets,whichmayberandomorintroducedbyanadversary, [16]) characterized orbit codes which result from the action
X
and D the transfer matrix with respect to these error packets. of cyclic subgroups. Recently, in [17], the classical tool of
r
a In non-coherent or random network coding, it is generally Plu¨cker coordinates was employed to compute the so-called
assumed that the source and the destination nodes have no ‘Plu¨ckerorbits’associatedwithanorbitcodeandthedefining
knowledge of the network topology - the only assumption equations of Schubert varieties were used in the description
is that the number of error packets, i.e. the parameter t, is of constant dimension subspace codes.
bounded. The contributions of this paper are listed below.
Themotivationforconstructionofsubspacecodesforrandom
(1) We have used some classical results ([18]) to provide a
networks over the projective space came from the seminal
unifiedframeworkforsomestandardconstructionsofsub-
work of Ko¨tter and Kschischang [1], where the problem of
space codes (e.g. in [1], [2], [3]) in terms of the Plu¨cker
error-and-erasure correction in random linear networks was
coordinatesoftheconstructedsubspaces.Inparticular,we
translated into that of transmission and recovery of linear
show that a strong correspondence exists: the generat-
subspacesofaprojectivespace.Subsequentresearch([2], [5])
ing matrices in both the so-called ‘lifting construction’
constructed subspace codes, with subspace distance ([1]) as
and its generalization correspond exactly to the matrix
the metric, basedon rank-metriccodes.A verysignificantde-
of the independent Plu¨cker coordinates with appropriate
velopmentin theconstructionofconstantdimensionsubspace
constraints on the choice of indices.
codes was the use of Ferrers diagram representation of rank-
metric codes which fit into a row-reduced echelon form rep- (2) We show that the characteristic representation of a Schu-
resentation of subspaces in a Grassmannian ([3]), which was bert cell in a projective space in terms of the indices
corresponding to a non-zero Plu¨cker coordinate can be B. The Plu¨cker Coordinates of Subspaces
used as an “identifying vector”([3], [4], [11]) for sub- A d-dimensional linear space L in Pn is a set of points
d
spaces in that Schubert cell. In particular, we show that P =(p(0),··· ,p(n)) ∈ Pn whose coordinatesp(j) satisfy a
both the subspace distance and injection distance among
system of n−d independent linear equations:
the subspacesin a projectivespace can be boundedusing
n
the subset distance among the defining index sets of
α p(j) = 0; i=1,··· ,n−d.
ij
corresponding Schubert cells.
j=0
X
(3) We give a construction of non-constant dimension sub-
There exist d + 1 points P = (p (0),··· ,p (n)) ;i =
i i i
space codes which is an extension of a construction
0, ··· , d, which span the d-space L . Consider the (d +
d
in [13] to the non-constant dimension case - for both
1)× (n +1) matrix P = (p (j)); denote as D(j ,··· ,j )
i 0 d
subspace distance and injection distance. The codes are
the determinant of the (d+1)×(d+1) submatrix formed
describedasunionsofSchubertcellsselectedtoguarantee
by those columns of P indexed by the sequence of integers
a minimum subspace or injection distance. Moreover,we
j ≤ ··· ≤ j , 0 ≤ j ≤ n. It follows that at least one of
0 d k
show that the Khaleghi-Kschischang construction [11] the N+1 such determinants, where N := n+1 −1, must
for non-constant dimension Ferrers-diagram rank-metric d+1
be non-zero. Written in a lexicographic order on the column
subspacecodesissubsumedin ourproposedconstruction indices, (··· ,D(j ,··· ,j ),···) determines(cid:0)a p(cid:1)oint in PN;
0 d
if the constant dimension subcode within each Schubert
these determinants constitute the Plu¨cker coordinates of the
cell is a suitable rank-metric code.
d-dimensional linear (projective) subspace L .
d
Theorganisationofthispaperisasfollows:SectionIIisa re- Theorem 2.2: ([18], [21]) There exists a natural bijective
vision of pertinent mathematicalresults regardingthe Plu¨cker correspondencebetween the d-spaces in Pn and the points of
coordinate description of projective subspaces and Schubert PN whose Plu¨cker coordinatessatisfy the following relations:
cells. In Section III, we reformulate standard subspace code
d+1
constructionsin the language of Plu¨cker coordinatesand give (−1)αD(j0,··· ,jd−1,kα)D(k0,··· ,kˇα,··· ,kd+1)=0.
a unified framework. The next section (Section IV) gives
α=0
the general construction of non-constant dimension subspace X (1)
codes. We conclude with a summary of the results and a where kˇα denotes omission of kα and 0≤jβ, kγ ≤n.
discussion on the directions of ongoing and future research ItiseasilyverifiedthatthePlu¨ckercoordinatesofanyd-space
in this context. inPn indeedsatisfytheaboverelations.Provingtheconverse,
i.e. any point of PN whose coordinates satisfy the above
relations corresponds to a unique d-plane in Pn, offers some
II. PRELIMINARYCONCEPTS
interesting insights [18]. For instance, it can be shown that
any d-subspace of the projective space Pn has an associated
In this section we briefly review and organize some math-
matrix representation having the (d+1)×(d+1) identity
ematical concepts which will be used in interpreting recent
matrix as a sub-matrix. The following proposition spells this
results in random network code constructions and lead to the
out in detail.
formulation of new coding schemes.
Proposition 1: [18] There is a natural bijective correspon-
dence between the set of points of PN whose coordinates
A. Projective Space over a Field
satisfy the quadraticrelationsin Theorem2.2, with the condi-
tionD(k ,··· ,k )6=0,andtheaffine(d+1)(n−d)spaceof
We begin with the definition of a projective space over an 0 d
(d+1)×(n+1)matrices(p (j)), i=0,··· ,d, j =0,··· ,n,
arbitrary field. i
such that the (d + 1) × (d + 1) sub-matrix p (k ), i =
Definition 2.1: [19] The projective space associated with a i α
vector space V over a field K is the set P(V) of lines in V. 0,··· ,d, α = 0,··· ,d is the identity. Moreover, such a
matrix p (j) correspondsto the point of PN with coordinates
Moreover, the projective space P(V) is associated with the i
D(j ,··· ,j )=Det(p (j ));suchapointof PN corresponds
K-vector space V in the following canonical way [19]: 0 d i β
to a (d+1)×(n+1) matrix with entries:
Considertheequivalencerelation ∼onthesetV∗ :=V \{0},
where, for x,y ∈ V∗, x ∼ y if and only if y = ax, a ∈ pi(j)=D(k0,··· ,ki−1,j,ki+1,··· ,kd)/D(k0,··· ,kd).
K, a 6= 0. Then the canonical map P : V∗ 7−→ P(V) (2)
associates each vector x with the projective point Kx. It follows from the proposition that the set of points of PN,
The dimension of P(V) is defined as Dim(P(V)) := whose co-ordinates satisfy the quadratic relations, is covered
Dim(V) − 1; P(0) is empty, and its dimension is −1. by N + 1 copies of the affine (d + 1)(n − d) space; this
Therefore, a d-dimensional subspace or a d-space of the n- is the Grassmann manifold or the Grassmannian denoted by
dimensionalprojectivespace Pn(K) overa field K is a setof G(d+1,n+1). In the next section we show that the above
pointswhoserepresentativevectorsalongwiththezerovector, proposition is the key to the interpretation of some well-
form a d+1-dimensional subspace of the n+1-dimensional knownconstructionsforconstantdimensioncodesforrandom
K-vector space V, where Pn(K)=P(V) [20]. networks in terms of the Plu¨cker coordinates of subspaces.
Henceforth we term the matrix (p (j)) as the Plu¨cker coordi- Definition 3.1: [2] Let X be an l×m matrix over F . We
i q
nate matrix of the subspace whose representation as a point define the ‘lifting’ map as follows:
of PN is given by: (··· ,p (j),···) in lexicographic order.
i I:Fl×m −→G(l,l+m); X 7−→I(X)=h[I X]i
q
C. Schubert Cells and their Representation
where I denotes the l ×l identity matrix and hMi denotes
Let G(d,n) be the Grassmannian of d-dimensional sub- the row-space of the matrix M. The subspace I(X) is called
spacesof an n-dimensionalK-vectorspace. For U ∈G(d,n), a ‘lifting’ of the matrix X.
thereexistsauniquelydeterminedbasis{v1,v2,··· ,vd}, vi ∈ IfC isarank-metriccode(e.g.[23]),thesubspacecodeI(C)=
Kn, of the form: {I(C)|C ∈C} is called the ‘lifting’ of C.
v =(∗,··· ,∗,1,0,··· ,0), A generalization of the lifting construction was achieved in
1
v =(∗,··· ,∗,0,∗,··· ,∗,1,0,··· ,0), [3] and [11], wherematricesin the row reducedechelonform
2
.
. (RREF) have been used in conjunction with Ferrers diagram
.
representation of rank-metric codes to construct the desired
v =(∗,··· ,∗,0,∗,··· ,∗,0,∗,··· ,∗,1,0,··· ,0).
d
subspaces.Wenowbrieflystatethisconstructionthroughsome
(d−1)zeros relevant definitions ([3], [11]).
The subspace U is uniquely determined by the placement
| {z } Definition 3.2: A Ferrers diagram [22] is a graphical
of the terminal 1’s in the v ’s, indexed by a sequence of
i representation of integer partitions as an array of dots such
integers 1 ≤ b < b < ··· < b ≤ n, and the elements
1 2 d that the i-th row has exactly the same number of dots as the
of K which occupy the ∗ positions. All the subspaces which
i-thelementinthepartition.Anm×n Ferrersdiagramhas m
are parametrized by a particular tuple β = (b ,b ,··· ,b ),
1 2 d dots in the rightmost column and n dots in the topmost row.
denotingthepositionsofthetrailing1’sinabasisoftheabove
Definition 3.3: Let F be a l×m Ferrers diagram. A rank-
form,aresaidtoconstitutetheSchubertCellS ofdimension:
β metric code C is called a Ferrers diagram rank-metric code
d (FDRMcode)associatedwithF ifallthecodewordsarel×m
d+1
d =(b −1)+···+(b −d)= b − . matrices in which the positions of the non-zero entries match
β 1 d i
2
i=1 (cid:18) (cid:19) thoseofthedotsinF.Inthiscase,thecodewordmatricesare
X
said to fit the Ferrers diagram F.
AnalternativedescriptionofSchubertcellsinvolvessubspaces
Definition 3.4: Let v ∈{0,1}n+1 be a vector of weight k.
whose basis vectors form a d × n matrix in row reduced
The profile matrix of v, denoted P , is a k×(n+1) matrix
echelonform,with the definingd-tuple indexingthe positions v
in RREF such that:
of the leading 1’s instead. The Schubert cell constituted of
such subspaces, say, S with α=(a ,a ,··· ,a ) and the a (i) The leading coefficients of the rows of P appear in the
α 1 2 d i v
satisfy 1≤a <a <···<a ≤n, has dimension: columns indexed by supp(v).
1 2 d
(ii) The entries in P other than 0’s and 1’s are dots.
d = (n−a −(d−1))+···+(n−a ) v
α 1 d
Extracting the dots of P forms a Ferrers diagram which is
d v
d
= nd− a − . denoted by F(v).
i
2 Definition 3.5: Let v ∈ {0,1}n+1 be a vector of weight
i=1 (cid:18) (cid:19)
X k and let X ∈ Fl×m, l ≤ k, m ≤ n−k +1, be a matrix
Inboththecases,themaximumdimensionn(n−d)isachieved q
which fits F(v). Then the (generalized)lifting of X (by v) is
for the tuple (1,··· ,d) and the minimum dimension is 0 for
defined as I (X)=hP (X)i⊆G(l,l+m), where P (X) is
v v v
the tuple (n−d+1,··· ,n). The second characterization of
the profile matrix of v with the dots replaced by the entries
Schubert cells is relevant to the construction of non-constant
of X.
dimension subspace codes for random networks.
As before, if C is a rank-metric code, then I (C) =
v
{I (C)|C ∈C} is called the generalized‘lifting’of C, which
III. A UNIFIEDFRAMEWORK FORLIFTING v
is an FDRM subspace code associated with F(v).
CONSTRUCTIONS USINGPLU¨CKERCOORDINATES
In this section, we reformulate two well-known methods B. Unified Framework Using Plu¨cker Coordinates
forconstructingconstantdimensionalsubspacecodesinterms Tointerprettheaboveconstructionsofconstantdimensional
of the Plu¨cker coordinates of certain subspaces, providing a subspace codes in terms of the Plu¨cker coordinates of the
unified framework for both of them. associated subspaces, we provethe followingpropositionas a
consequence of Theorem 2.2 and Proposition 1.
A. Overview of the Lifting Constructions Proposition 2: If X ∈ F(d+1)×(n+1) is in row-reduced
q
The Lifting Construction was first proposed in [1] to yield echelon form (RREF), then the Plu¨cker coordinate matrix
constant dimensional Reed-Solomon-like codes for error and associated with the subspacehXi⊆G(d+1,n+1)coincides
erasure correction in random networks. In a subsequent work with X, upto multiplication by a unit.
[2], constant dimension subspace codes were constructed by Proof: Suppose the columns k ,k ,··· ,k of X contain the
0 1 d
‘lifting’ rank-metric codes as follows. leading elements of the rows. As D(k ,··· ,k ) 6= 0, by
0 d
Proposition 1, hXi is identified with a point of PN, where is:
N= n+1 −1, corresponding to a (d+1)×(n+1) matrix 1 0 1 0 1 0
d+1
with entries: 0 1 1 0 1 0
(cid:0) (cid:1) Iv(X)=0 0 0 1 1 0
pi(j)=D(k0,··· ,ki−1,j,ki+1,··· ,kd)/D(k0,··· ,kd), 0 0 0 0 0 1
 
Thepositionsofthenon-zeroentriesinv areindexedas:k =
which is the associated Plu¨cker coordinate matrix. 0
0,k =1,k =3,k =5. Hence the matrix corresponding to
Now we give the following interpretations of the lifting 1 2 3
thePlu¨ckercoordinateD(0,1,3,5)isthe4×4identitymatrix
construction and its generalization in terms of the Plu¨cker
and so, D(0,1,3,5) = 1. As k−l = 1 and m = n−k+1,
coordinates.
boththematrixX′ andtheaugmentedmatrixA(X)aregiven
Lifting Construction: In the setting of Proposition 2, set
by:
d = l −1,n = l +m−1 and 0 ≤ k < ··· < k ≤ n,
0 d 1 1
with D(k ,··· ,k ) 6= 0. Let X ∈ Fl×m, for instance, a
0 d q 1 1
codewordmatrixofanl×mrank-metriccodeoverFq.Selecta X′ =A(X)=0 1
subspaceofG(d+1,n+1)whosePlu¨ckercoordinatessatisfy
0 0
the following conditions: k = i,i = 0,··· ,d, i.e. we have  
i
 
D(0,··· ,l −1) = 1 and all other coordinates are obtained Consider the four coordinates: D(2,1,3,5),D(0,2,3,5),
from those of the form D(j ,··· ,j ), 0 ≤ j ≤ n, at most D(0,1,2,5),D(0,1,3,2),involvingthe first column of A(X)
0 d β
one of the j ∈/ {0,··· ,l −1}, and that particular column arising out of the column of P with index 2. We have:
β v
is from the matrix X in ascending order. This subspace is
1 0 0 0 1 1 0 0
precisely the lifted subspace I(X)=h[I X]i.
1 1 0 0 0 1 0 0
GeneralizedLiftingConstruction:Thegeneralizedliftingcon- Det =1, Det =1
0 0 1 0 0 0 1 0
struction may similarly be described as the selection of sub-
0 0 0 1 0 0 0 1
spaceswithparticularconstraintsontheirPlu¨ckercoordinates.    
,    
Given a vector v as described, set k = d + 1 and let
k , i=0,··· ,d, 0≤k ≤n,denotethepositionsofnon-zero 1 0 1 0 1 0 0 1
i i
componentsofv inascendingorder.If X ∈Fl×m beamatrix 0 1 1 0 0 1 0 1
q Det =0, Det =0.
which fits F(v), where l ≤ k, m ≤ n−k+1, (e.g. X is a 0 0 0 0 0 0 1 0
codewordofaFDRMcodeassociatedwiththeFerrersdiagram 0 0 0 1 0 0 0 0
   
F(v)) construct the augmented matrix A(X)∈Fk×n−k+1 in    
q which is in accordance with Proposition 2; and the Plu¨cker
two steps as follows:
coordinate matrix coincides with the lifted matrix I (X).
v
(1) To each column of X append k−l zeros from below to It follows that both the lifting construction and its general-
form X′ ∈Fk×m. ization have precise interpretations in terms of the Plu¨cker
q
(2) Add n−k−m+1 all-zerocolumnsamongthe columns coordinatedescription of subspaces. In fact this interpretation
ofX′preservingtheorderofthosecolumnsin P toform provides a unified framework as the encoding procedure is
v
A(X). shown to be equivalentto a choice of a set of column indices
from the underlying (codeword) matrices and computing the
SelectasubspaceofG(d+1,n+1)whosePlu¨ckercoordinates
Plu¨cker coordinates for those particular indices. In the lifting
satisfy the following constraints: D(k ,··· ,k ) = 1 and the
0 d construction the ‘pivotal’ set of indices {k ,··· ,k } was
remaining coordinates are obtained from those of the form 0 d
constrained to be just the set {0,··· ,d}. Hence it actually
D(j ,··· ,j ), 0 ≤ j ≤ n, where at most one of the
0 d β limits the choice of subspaces to the so-called principal
j ∈/ {k ,··· ,k }, and that particular column is from the
β 0 d Schubert cell, with a corresponding matrix representation
augmented matrix A(X) in ascending order. This subspace is
having an identity matrix in the leftmost d+1 columns. In
the lifted subspace I (X)=hP (X)i.
v v the generalized lifting construction, the (d + 1) × (d + 1)
We now give a simple example illustrating the Plu¨cker coor-
identity sub-matrix is interspersed with columns from the
dinate description of the generalized lifting construction.
underlying (codeword) matrix; hence, other Schubert cells
Example Let v = 110101∈ {0,1}6, i.e. n = 5,k = 4. Then
are also exploited. But, in both cases the lifted matrix still
the profile matrix of v and the associated Ferrers diagram are
coincides with the (d+1)×(n+1) matrix formed by the
given by:
‘independent’ Plu¨cker coordinates in the setting of Theorem
2.2 and Proposition 1, as has been outlined earlier.
1 0 • 0 • 0
• •
0 1 • 0 • 0
Pv =0 0 0 1 • 0 F(v)=• • IV. A CONSTRUCTION OFNON-CONSTANTDIMENSIONAL
• SUBSPACE CODES
0 0 0 0 0 1
  The construction of non-constant dimensional subspace
 
In this case, l =3<k =4, but m=2=n−k+1. Now if codes has been addressed in [3] and [11]: in the former,
X ∈F3×2 isamatrixwhich‘fits’ F(v),thentheliftedmatrix by ‘puncturing’ a constant dimension code that results from
2
a multilevel code construction; in the latter, by partitioning (1) Selection of Schubert cells whose constituent subspaces
the projective space followed by a two-level selection of across the cells are at a prescribed minimum subspace
subspaces. In this section we present a construction of non- distance from each other;
constant dimension subspace codes which is an extension of (2) Selection of subsets of subspaces in each Schubert cell
the construction of orbit codes for random networks [13] for which have the prescribed minimum subspace distance
constant dimension codes. The Khaleghi-Kschischang con- among them.
struction [11] also employs the so-called injection distance
In [11], the selection of subsets within a Schubert cell is
metric,ratherthanthesubspacedistancemetric,astheformer
achieved via lifting of Ferrers diagram rank-metric (FDRM)
sometimes outperforms the latter in a worst-case adversarial
codes, while a profile vector selection algorithm is imple-
model. So we first outline our construction for subspace
mented to choose appropriate Schubert cells which also sup-
distance and then discuss the modifications required with
portthelargestliftedFDRMcodes.Asdiscussedinpreceding
injection distance as the metric.
sections, Schubert cells can be characterized by the tuples
which indexthe positionsof leading1’s of a matrix in RREF.
V. CODES WITHSUBSPACE DISTANCE ASMETRIC
Following Proposition 3, the selection of a Schubert cell is
First we describe the selection of subspaces in terms of translated into the selection of a tuple of column indices in
theirPlu¨ckercoordinates.RecallfromProposition1that every the Plu¨cker coordinate matrix of a representative subspace.
d-space in the projective n-space has a non-zero Plu¨cker For the selection of subspaces within a Schubert cell we
coordinate D(k ,··· ,k ) such that the (d + 1) × (d + 1) propose a method exploiting the characteristic representation
0 d
submatrix of the coordinate matrix formed by the columns ofsubspacesinthe Schubertcell.First westate some relevant
indexed by k , i=0,··· ,d, is the identity. Let U, V be two definitions.
i
different subspaces, of possibly different dimensions d , d , Definition 5.1: Let the set of basis vectors of any subspace
u v
with the tuples {ku} and {lv} as the column indices of the of a Schubert cell Sα have a d×n matrix representation in
identity submatrix in each case. Then the symmetric distance RREF with the positions of the leading 1’s indexed by the
between the tuples {ku} and {lv} givesa lowerboundon the tuple {α} = {a1,a2,··· ,ad}. The Schubert cell matrix Aα
subspace distance ds(U,V) as follows. corresponding to the Schubert cell Sα is the d×n matrix in
Proposition 3: Let U, V be subspaces of Pn of projective RREF in which the columnsindexedby the tuple {α} form a
dimensionssandtrespectively;andlet{k ,k ,··· ,k }, 0≤ d×d identity matrix and all other columns are zero.
0 1 s
kα ≤ n, and {l0,l1,··· ,lt}, 0 ≤ lβ ≤ n, be the tuples Definition 5.2: Given a Schubert cell Sα characterized by
specifying the identity submatrices of the respective Plu¨cker the tuple {α} = {a1,a2, ··· ,ad}, 1 ≤ a1 < ··· <
coordinate matrices. Then ds(U,V) ≥ ∆({kα},{lβ}), where ad ≤n, the complementary Schubert cell Sαc is characterized
∆({u},{v}) is the symmetric distance between the sets by the tuple {β} = {b1,··· ,b(n−d)} = {1,2,··· ,n} \
{u}, {v}. {a1,a2,··· ,ad}.
Proof: Without loss of generality assume that the Plu¨cker Evidently the Schubert cell matrix for Sαc is given by the
coordinate matrices M(U), M(V), of respective dimensions (n−d)×n matrix Acα in RREF with the columns indexed
(s+1)×(n+1)and (t+1)×(n+1),are in RREF, keeping by {β} forming an (n−d)×(n−d) identity matrix and the
the columns of the identity submatrices intact. Now remaining columns all zero.
Let S be a Schubert cell with the associated d×n Schubert
α
∆({kα},{lβ})=|{kα}\{lβ}|+|{lβ}\{kα}|. cell matrix Aα. Also denote the d×n RREF ‘matrix’ rep-
resentation of the S with asterisks indicating arbitrary field
α
Let a := |{k }\{l }|, b := |{l }\{k }|. Then there are
α β β α elements (vide Subsection II-C) as M(S ).
α
s + 1 − a = t + 1 − b matching positions in the matrices
Consider the set of d × (n − d) matrices such that the
M(U), M(V) where there are leading 1’s; hence we have
columns of these matrices correspond to those columns in-
Dim(U ∩V)≤(1/2)(s+t−(a+b)). But then
dexed by the set: β = {b1,b2,··· ,bn−d} := {1,2,··· ,n}\
{a ,a ,··· ,a }inascendingorder,havingnon-zeroelements
d (U,V) := DimU +DimV −2Dim(U ∩V) 1 2 d
s
only in the positions of the asterisks in M(S ). Select an
α
≥ s+t−2((1/2)(s+t−(a+b)))
additive group G of these matrices such that each element of
= a+b. the group has rank at least r/2.
For each G∈G construct the d×n matrix C(G) as follows:
TheaboveresultisageneralreformulationofLemma2in[3], C(G)=Aα+G·Acα.
which relates the subspace distance between the subspaces of The constantdimension subspace code constructedwithin the
a projective space with the Hamming distance between their Schubert cell S is then given by:
α
binary profile vectors.
C ={hC(G)i|G∈G}.
Following [11], a two-step strategy for constructing non- α
constant dimension subspace codes over a projective space The minimum subspace distance of the code satisfies d ≥
min
is outlined below: r, as borne out in the following theorem.
Theorem 5.3: Let G , G belong to an additive group of Definition 6.1: [10] The injection distance between sub-
1 2
d × (n − d) matrices with minimum rank r/2. Then the spaces U, V of Pn is given by:
subspaces spanned by the matrices C(G ) and C(G ) within
1 2
d (U,V):=max{DimU,DimV}−Dim(U ∩V)
the Schubert cell S are at a minimum subspace distance I
α
d ≥r, where C(G ), i=1,2, are defined as above. =Dim(U +V)−min{DimU,DimV}.
min i
Proof: The d×n matrices C = C(G ), i = 1,2, have the
i i The injection distance between two subspaces is thus related
matricesG , G ,whichbelongtotheadditivegroupandhave
1 2 to the subspace distance between them as follows ([10]):
minimumrankr/2, asthed×(n−d)submatricesconstituted
1 1
of the columns indexed by b1,b2,··· ,bn−d. Then we have: dI(U,V)= ds(U,V)+ |DimU −DimV| (3)
2 2
d (C , C )=2rank C1 −2d=2rank C1−C2 −2d≥r. It is evident that, upto a scaling factor, the injection distance
S 1 2 C2 C2 coincides with the subspace distance for constant-dimension
(cid:20) (cid:21) (cid:20) (cid:21)
subspace codes. It was observed in [10] that the injection
The last step follows from the fact that the non-zero columns
distance gives a more precise measure of the subspace code
of the difference matrix C −C constitute an element of the
1 2 performance when a single error packet can cause simultane-
additive group of matrices of minimum rank r/2.
ous error and erasure. However, in a less pessimistic channel
Hence we can describe our two-step construction of non-
modelthantheworst-caseadversarialchannel,forinstance,in
constantdimensionsubspacecodeswitha minimumsubspace
the case where a single error packet can introduce either an
distance d as follows.
min error or an erasure but not both, the subspace distance is still
(1) Selection of Schubert cells: Select the largest subset S of appropriate.
Schubertcells subjectto the followingcondition:Forany
B. ConstructionofNon-constantDimensionSubspaceCodes:
S ,S ∈S, we have ∆({α},{β})≥d ;
α β min
First we prove a counterpart of Proposition 3, with the
(2) Selection of subspaces in each Schubert cell: The sub-
same setting and notation, for the injection distance d (U,V)
spaces within a Schubert cell are chosen such that their I
between subspaces U, V of Pn.
spanning matrices satisfy the following condition: The
Proposition 4: Let U, V be subspaces of Pn of projective
submatrix indexed by the characteristic columns of the
dimensionssandtrespectively;andlet{k ,k ,··· ,k }, 0≤
complementary Schubert cell matrix belongs to an addi- 0 1 s
k ≤ n, and {l ,l ,··· ,l }, 0 ≤ l ≤ n, be the tuples
tive group of minimum rank ⌊d /2⌋. α 0 1 t β
min
specifying the identity submatrices of the respective Plu¨cker
The non-constant dimension subspace code is, therefore, the coordinate matrices. Then d (U,V) ≥ ⌊1∆({k },{l })⌋,
union: C , where the union is over all the Schubert cells I 2 α β
α where∆({u},{v})isthesymmetricdistancebetweenthesets
S ∈S.
α {u}, {v}.
S
Proof: From the proof of Proposition 3, we obtain:
VI. CODES WITHINJECTIONDISTANCE ASMETRIC Dim(U∩V)≤(1/2)(s+t−(a+b)),wherea:=|{k }\{l }|
α β
and b:=|{l }\{k }|. So we have:
β α
Inthissectionwefirstintroducethenotionofinjectiondis-
tance anddiscussits applicabilityvis-a`-vissubspacedistance. 2d (U,V) := 2max{DimU,DimV}−2Dim(U ∩V)
I
Next we modify the construction of non-constant dimension
≥ DimU +DimV −2Dim(U ∩V)
subspace codes presented earlier to adapt it for the injection
≥ a+b.
distance metric.
As ∆({k },{l })=a+b, the proposition follows.
α β
A. The Adversarial Channel Model and Injection Distance Alternatively, from Equation 3 and Proposition 3, we have:
between Subspaces:
2d (U,V)≥∆({k },{l })+|s−t|
I α β
Recallthatinthelinearmulticastnetworkcodingmodelthe
and the proposition follows.
received packets at a destination are expressed as the rows of
As suggested above, to capture the effect of the injection
a matrix: Y =AX+DZ, where the matrix A is the network
distancemetricin ourframework,we modifythedefinitionof
transfermatrixseen bythesourcepackets(rowsof X)andD
the symmetric distance between two sets.
isthetransfermatrixseenbytheerrorpackets(rowsof Z).In
Definition 6.2: For two finite sets S ,S of respective
the general random network coding problem, the matrix A is 1 2
cardinalitiess ,s ,themodifiedsymmetricdistanceisdefined
unknownto the receiverexceptfora lowerboundon the rank 1 2
as follows:
ofthematrix.Intheworst-caseadversarialmodelinvestigated
in[10],thefollowingadditionalassumptionsaremade: ∆ (S ,S )=#{S \S }+#{S \S }+|s −s |
m 1 2 1 2 2 1 1 2
(1) The adversary knows the matrix X and can choose the
where#S denotesthecardinalityofthesetS and‘||’denotes
transfer matrix A while respecting the rank constraint;
the absolute value.
(2) The matrix D is arbitrarily chosen by the adversary. Proposition5:Themodifiedsymmetricdistanceisametric.
Proof: Itisevidentthat∆ (S ,S )=∆(S ,S )+|s −s |. where δ is the minimum rank distance of C.
m 1 2 1 2 1 2 r
Symmetry and the fact that ∆ (S ,S ) ≥ 0 with equality if Define V := F d×(n−d). The kernel of the surjective map:
m 1 2 q
and only if S = S follow immediately. Triangle inequality π : V → V/C is a subspace of dimension d(n −d) − D.
1 2
is satisfied because ∆() is a metric and for positive integers The largest FDRM code which matches M(S ) is the largest
α
s ,s ,s we have: |s −s |+|s −s |≥|s −s |. subcode C′ ⊆C subject to the constraint that all its codeword
1 2 3 1 2 1 3 2 3
Nowwereformulatethetwo-stepconstructionofnon-constant matriceshavezerosin d(n−d)−D fixedpositions.Invoking
α
dimension subspace codes with injection distance as the the standard vector-space isomorphism between F m×n and
q
metric. From Propositions 4 and 5, the selection criterion F mn where m,n are positive integers, any codeword c ∈C′
q
of Schubert cells for the injection distance metric is read- may be expressed as a d(n−d)-dimensional vector over F
q
ily obtained from the subspace distance case. Moreover, as with non-zero elements in D coordinates i ,··· ,i . Let
α 1 Dα
the second step is essentially a construction of a constant- φ:V →F d(n−d)−Dα be the surjective map which sets zero
q
dimensionsubspacecodewithinaSchubertcell,thetransition the coordinates i ,··· ,i . Now consider the linear map:
1 Dα
from subspace distance to injection distance involves only a
Φ:V −→V/C×F d(n−d)−Dα; v 7−→(π(v),φ(v)).
scaling factor. Hence we may state the construction of non- q
constantdimensionsubspace codeswith a minimuminjection Evidently, DimΦ(V) ≤ 2d(n −d)−D −D . The largest
α
distance δmin as follows. FDRM code is the kernel of the map Φ, and hence, by the
(1) Selection of Schubert cells: Select the largest subset S of rank-nullity theorem, has dimension:
Schubertcells subjectto the followingcondition:Forany
κ≥d(n−d)−(2d(n−d)−D−D )
α
S ,S ∈S, ∆ ({α},{β})≥2δ ;
α β m min =D +max{d,n−d}(min{d,n−d}−δ +1)−d(n−d)
α r
(2) Selection of subspaces in each Schubert cell: The sub- =D −max{d,n−d}(δ −1).
α r
spaces within a Schubert cell are chosen such that their
spanning matrices satisfy the following condition: The
submatrix indexed by the characteristic columns of the A. Selection of Schubert Cells:
complementary Schubert cell matrix is an element of an
It follows from Theorem 7.1 that the size of the FDRM
additive group of minimum rank δ .
min non-constant dimension subspace codes depends directly on
Thenon-constantdimensionsubspacecodeistheunion: Cα, the dimension of the chosen Schubert cells. In [11] the
over all the Schubert cells Sα ∈S, as defined earlier. relation between the so-called profile vectors associated with
S
each Schubert cell and the dimension of the FDRM codes
VII. NON-CONSTANTDIMENSION FERRERS-DIAGRAM
constructedhasbeen used in an algorithmforselection of the
RANK-METRICSUBSPACE CODES
Schubertcells. We presenta moredirectapproach,basingour
In the proposed construction of non-constant dimension
choice on the dimension of each Schubert cell, as follows.
subspace codes, if the additive group G of d × (n − d)
Recall from Subsection II-C that the dimension of a Schubert
matrices is chosen to be a rank-metric code [23], we have a
cellS inthesecondrepresentationcharacterizedbythetuple
α
so-callednon-constantdimensionFerrers-diagramrank-metric {α} is given by: D = nd(α)− d(α)a − d(α) , where
code (FDRM) subspace code. We show that the Khaleghi- α i=1 i 2
d(α) := |{α}| = d is the length of the tuple. Let A denote
Kschischang construction [11], which has resulted in some P (cid:0) (cid:1)
theset ofallsuch distincttuplesα inlexicographicorder.For
of the best-known subspace codes in terms of code rate, is
each α∈A define the choice function:
subsumed in our framework when the additive group G is a
rank-metric code. F(S )=D −max{d(α),n−d(α)}(δ −1). (4)
α α r
In the proposed construction, the FDRM codeword matrices
The selection of Schubert cells proceeds via a greedy algo-
havenon-zeroelementsinpositionswhichmatchtheasterisks
rithm, similar to that in [11], which selects at each stage the
in the ‘matrix’ M(S ), where S is the Schubertcell charac-
α α tuple {α} with the largest F(S ) and discards all the tuples
terizedbythe tuple {α}={a ,a , ··· ,a }.First we provea α
1 2 d β ∈ A such that ∆({α},{β}) < d for the subspace dis-
lowerboundonthedimensionofsuchanFDRMcode,similar min
tancemetricor⌊1∆({α},{β})⌋+⌊1|#{α}−#{β}|⌋<δ
to Theorem 3 in [11], in this general setting. 2 2 min
for the injection distance metric. The next stage proceeds
Theorem 7.1: The dimension κ of the largest FDRM code
by applying the same choice function to the modified set
whose codewords have non-zero elements precisely in the
A:=A\B, where B is the set of the discarded tuples {β} at
positions of the asterisks in M(S ) satisfies: κ ≥ D −
α α the previous stage. The selection algorithm terminates when
max{d,n−d}(δ −1),whereδ istheminimumrankdistance
r r A:=∅.ThechoicefunctioninEquation4,whichis basedon
of the rank-metric code and D , the number of asterisks in
α thelowerboundonthedimensionoftheFDRMsubspacecode
M(S ), is the dimension of the Schubert cell S .
α α given in Theorem 7.1, ensures that at every stage a Schubert
Proof: It is known [23] that the dimension of a linear
cell which supports the largest FDRM code is chosen among
maximumrank-distance(MRD)codeC,whosecodewordsare
the contendingcells. The largest numberof subspaces belong
d×(n−d) matrices over F , is given by:
q to the principal Schubert cell of the Grassmannian G(d,n),
D =max{d,n−d}(min{d,n−d}−δ +1) with d = ⌊n/2⌋ or d = ⌈n/2⌉; for these values of d, the
r
negative term in the choice function is also minimum for a q dI ds N LB(ds) KK(ds) LB(dI) KK(dI))
2 2 4 9 15.1515 15.1518 15.3238 15.6245
given δr. Hence the algorithm proceedsby choosing either of 2 2 4 10 20.1534 20.1534 20.1967 20.3294
these subspaces (different when n odd) in the first step. 2 2 4 12 30.1556 30.1557 30.1998 30.3346
2 3 6 13 28.0030 28.0030 28.0134 28.0263
3 2 4 7 8.0131 8.0145 8.0464 8.1331
B. Comparison with the Khaleghi-Kschischang algorithm:
3 2 4 8 12.0135 12.0135 12.0160 12.0311
The above algorithm and that stated in [11] differ in 4 2 4 7 8.0030 8.0032 8.0142 8.0522
4 2 4 8 12.0031 12.0031 12.0034 12.0068
two aspects, one of which can be explained as a general
TABLEI
reformulation and the other, a slight modification.
COMPARISONOFLOWERBOUNDSONTHERATESOFOURFDRM
(i) Binary Profile Vectors vs. Characteristic Tuples: In [11], SUBSPACECODESWITHTHEACTUALRATESOFCODESIN[11]
the selection of Schubert cells is based on the so-called
binary profile vectors associated with each cell. The
Ferrersdiagramrepresentationof eachprofile vectorthen
It is seen that the calculated lower bound is not simply
gives the dimension and the positions of non-zeroentries
a computation of the choice function in each case and then
of the appropriate rank-metric code. In our construction
summing the values in the exponent.The justification for this
we simply use the characteristic tuple representation of
modification is as follows: The choice function F(S ) for
α
each Schubert cell and the rank-metric code is likewise
a Schubert cell S ∈ S is zero or negative when it does
α
specifiedbytheso-calledSchubertcell“matrix”(withthe
not support a rank-metric code of the given dimensions with
asterisks in the “basis vectors” replacing the dots in the
theprescribedminimumrank-distance.However,theparticular
Ferrers diagrams) - a direct formulation.
Schubert cell still maintains the inter-cell minimum subspace
(ii) The Choice Function for the Greedy Algorithm: The or injection distance and so contributes at least one subspace
following scoring function is used to select a new profile codeword to the entire code. To give a simple illustration,
vector v in [11]: consider the following construction of a non-constant dimen-
sionFDRMsubspacecodeinP4(F )withminimumsubspace
n i 2
score(v,δ )= v¯v −max{wt(v),η(v)}(δ −1), distance 4. A set S for this code consists of Schubert cells
r i j r
specified by the tuples {1,2},{3,4},{1,2,3,4,5}, with cor-
i=1j=1
XX
(5) responding choice function values of 3,−1,−5, respectively.
where η(v) = n−(wt(v)+mint∈s(v)t)+1, wt(v) the ForthesecondtupleM(Sα)hasonly2asteriskswhilethelast
(Hamming) weight of the binary vector v ∈ {0,1}n, one has none, being the 5×5 identity matrix, obviously not
v¯ the complement of v and s(v) the support of v, enoughto supportthe requiredrank-metriccodein each case.
i.e. the set whose elements denote the positions of the However,theycanstill contributeasingle subspacecodeword
1’s in v. It is evident that the sum term in the above each.
scoring function computes D , the number of asterisks InTable1wehavegivenafewlowerboundsonthecoderates
α
in the ‘matrix’ representation of the chosen Schubert in our construction with respect to given minimum subspace
cell in our framework, which, in our choice function distance (column LB(d )) and minimum injection distance
s
(Equation 4), is the exact formula based on the char- (column LB(d )) for some small field sizes q and projective
I
acteristic tuple. Rewriting the second term in Equation dimensions N −1. It is seen that they are comparable with
4 as max{d(α),n − (d(α) + min {a }) + 1}(δ − 1) those of the actual codes constructed in [11], which are also
i i r
makes it equivalent to the second term of Equation 5. presented in the columns ‘ KK(d )’ (minimum injection dis-
I
It modifies the parameter n − d(α) by subtracting the tance)and‘KK(d )’(minimumsubspacedistance).Infact,for
s
numberof leftmost all-zero columns in M(S ), which is subspace distance the rates of the constructed codes actually
α
not necessary for making a choice in our framework. coincide with the computed lower bound in half of the cases,
while the other values show only marginalimprovement.The
C. Numerical Results: difference between the rates of the constructed codes and
Wecomputelowerboundsontheratesofcodesconstructed the computed lower bounds is more pronounced for injection
for a given minimum subspace distance (using symmetric distance.
distance) and for a given minimum injection distance (using
the modified symmetric distance), from the values of the VIII. CONCLUSION
choice function (Equation 4) for each selected Schubert We have established a unified framework for the construc-
cell. For a selected set of Schubert cells S, a lower bound tion of constant-dimensionrandomnetworkcodes in terms of
on the code rate of a non-constant dimension FDRM theclassicaldescriptionofthesubspacesofa projectivespace
subspace code over a finite field Fq is given by the formula: usingthePlu¨ckercoordinates.Furtherwehavegivenageneral
logq( Sα∈SqF(Sα)) where constructionofnon-constantdimensionsubspacecodes,which
exploits a basic representation of Schubert systems, for both
P
F(Sα), if F(Sα)≥0 the subspace distance and the injection distance metrics. We
F(S )=
α (0 otherwise. havedemonstratedthatourframeworksubsumestheconstruc-
tion of [11], which is corroborated by the proximity of the [4] N. Silberstein and T. Etzion, “Enumerative Coding for Grassmannian
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ACKNOWLEDGMENT
The authoris gratefulto Dilip P. Patil, Sumanta Mukherjee
and Smarajit Das for their assistance.
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