Finite field matrix channels for network coding

0.3 MB·

by  Simon R. Blackburn

Finite Field Matrix Channels for Network
Coding
Simon R. Blackburn and Jessica Claridge
6 Department of Mathematics
1
Royal Holloway University of London
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2 Egham, Surrey TW20 0EX, United Kingdom

c
e
D
Abstract

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In2010,Silva,KschischangandK¨otterstudiedcertainclassesoffinitefieldmatrix

] channels in order to model random linear network coding where exactly t random
T
errors are introduced.
I
. In this paper we introduce a generalisation of these matrix channels that al-
s
c lows the modelling of channels with a varying number of errors. We show that a
[ capacity-achieving input distribution can always be taken to have a very restricted

form (the distribution should be uniform given the rank of the input matrix). This
2
v result complements, and is inspired by, a paper of Nobrega, Silva and Uchoa-Filho,
7 that establishes a similar result for a class of matrix channels that model network
3 coding with link erasures. Our result shows that the capacity of our channels can
0
be expressed as a maximisation over probability distributions on the set of possible
6
ranks of input matrices: a set of linear rather than exponential size.
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.
1
0 1 Introduction
6
1
Network coding, first defined in [1], allows intermediate nodes of a network to com-
:
v pute with and modify data, as opposed to the traditional view of nodes as ‘on/off’
i
X switches. This can increase the rate of information flow through a network. It is
shown in [8] that linear network coding is sufficient to maximise information flow in
r
a multicast problems, that is when there is one source node and information is to be
transmitted to a set of sink nodes. Moreover,in [6] it is shownthat for generalmul-
tisource multicast problems, random linear network coding achieves capacity with
probability exponentially approaching 1 with the code length.
In random linear network coding, the source injects packets into the network;
thesepacketscanbethoughtofasvectorsoflengthmwithentriesinafinitefieldFq
(where q is a fixed power of a prime). The packets flow through a network of un-
known topology to a sink node. Each intermediate node forwards packets that are
random linear combinations of the packets it has received. A sink node attempts to
reconstruct the message from these packets. In this context, Silva, Kschischang and
K¨otter[13]studied achanneldefinedas follows. We write Fn×m to denote the setof
q
all n×m matrices over Fq, and write GL(n,q) for the set of all invertible matrices
in Fn×n.
q
Definition 1. The Multiplicative Matrix Channel (MMC) has input set X and out-
put set Y, where X =Y =Fn×m. The channel law is
q
Y =AX
where A∈GL(n,q) is chosen uniformly at random.
1
Here the rows of X correspond to the packets transmitted by the source, the
rowsofY arethe packetsreceivedbythe sink,andthe matrixAcorrespondsto the
linear combinations of packets computed by the intermediate nodes.
Inspired by Montanari and Urbanke [9], Silva et al modelled the introduction of
random errors into the network by considering the following generalisation of the
MMC. We write Fn×m,r for the set of all n×m matrices of rank r.
q
Definition 2. The Additive Multiplicative Matrix Channel with t errors (AMMC)
has input set X and output set Y, where X =Y =Fn×m. The channel law is
q
Y =A(X +W)
where A ∈ GL(n,q) and W ∈ Fn×m,t are chosen uniformly and independently at
q
random.
So the matrix W corresponds to the assumption that exactly t linearly indepen-
dent random errors have been introduced. The MMC is exactly the AMMC with
zero errors.
We note that the AMMC is very different from the error model studied in the
well-known paper by K¨otter and Kschischang [7], where the errors are assumed to
be adversarial(so the worst case is studied).
In [13] the authors give upper and lower bounds on the capacity of the AMMC,
which are shown to converge in certain interesting limiting cases. The exact ca-
pacity of the AMMC for fixed parameter choices is hard to determine due to the
many degrees of freedom involved: the naive formula maximises over a probability
distributiononthe setofpossible input matrices,andthis setis exponentially large.
In this paper we introduce a generalisation of these matrix channels that allows
the modelling of channels were the number of errors is not necessarily fixed. (For
example,itenablesthe modellingofsituationswhenatmostterrorsareintroduced,
or when the errors are not necessarily linearly independent, or both.) To define our
generalisation, we need the following notation which is due to Nobrega, Silva and
Uchoa-Filho [10].
Definition 3. Let R be a probability distribution on the set {0,1,...,min{m,n}}
of possible ranks of matrices M ∈ Fn×m. We define a distribution on the set Fn×m
q q
ofmatrices bychoosingr accordingto R,andthen oncer is fixedchoosinga matrix
M ∈ Fn×m,r uniformly at random. We say that this distribution is Uniform Given
q
Rank (UGR) with rank distribution R. We say a distribution on Fn×m is Uniform
q
Given Rank (UGR) if it is UGR with rank distribution R for some distribution R.
We write R(r) for the probability of rank r under the distribution R. So a
distributiononFn×m isUGRwithrankdistributionRifandonlyifeachM ∈Fn×m
q q
of rank r is chosen with probability R(r)/|Fn×m,r|.
q
Definition 4. Let R be a probability distribution on the set {0,1,...,min{m,n}}
of possible ranks of matrices M ∈ Fn×m. The Generalised Additive Multiplicative
q
MAtrixChannelwithrankerrordistributionR(theGammachannelΓ(R))hasinput
set X and output set Y, where X =Y =Fn×m. The channel law is
q
Y =A(X +W)
where A ∈ GL(n,q) is chosen uniformly, where W ∈ Fn×m is UGR with rank
q
distribution R, and where A and W are chosen independently.
We see that the AMMC is the special case of Γ(R) when R is the distribution
choosing rank t with probability 1. But the model covers several other very natural
situations. For example, we may drop the assumption that the t errors are linearly
independent by defining R(r) to be the probability that t vectors span a subspace
of dimension r, when the vectors are chosen uniformly and independently. We can
also extend this to model situations when the number t of vectors varies according
2
to natural distributions such as the binomial distribution (which arises when, for
example, a packet is corrupted with some fixed non-zero probability). In practice,
givena particular network,one may runtests on the network to see the actualerror
patterns produced and define an empirical distribution on ranks. One could also
defineanappropriatedistributionbyconsideringsomecombinationofthe situations
described.
Throughoutthispaper,weassumethatq,n,mandRarefixedbytheapplication.
We will refer to these values as the channel parameters.
We note that the Gamma channel assumes that the transfer matrix A is always
invertible. Thisisarealisticassumptioninrandomlinearnetworkcodinginstandard
situations: thefieldsizeqisnormallylarge,whichmeanslineardependenciesbetween
random vectors are less likely.
In both [10]and[12]the authorsconsider (different) generalisationsofthe MMC
channelthatdonotnecessarilyhaveasquarefullranktransfermatrix. Suchchannels
allow modelling of network coding when no erroneous packets are injected into the
network, but there may be link erasures. In [10], Nobrega, Silva and Uchoa-Filho
define the transfer matrix to be picked from a UGR distribution. One result of [10]
is that a capacity-achieving input distribution for their class of MMC channels can
always be taken to be UGR.
A main result of this paper (Theorem 16) is that a capacity-achieving input
distribution for Gamma channels can always be taken to be UGR. Theorem 16 is a
significant extension of the result of [10] to a new class of channels; the extension
requires new technical ideas.
Corollary 18 to the main result of the paper provides an explicit method for
computing the capacity of Gamma channels which maximises over a probability
distributionoverthesetofpossibleranksofinputmatrices,ratherthanthesetofall
input matrices itself. Thus we have reduced the problem of computing the capacity
of a Gamma channel to an explicit maximisation over a set of variables of linear
rather than exponential size.
The remainder of the paper is organised as follows. Section 2 proves some pre-
liminary results needed in what follows. In Section 3 we state results from matrix
theory that we use. Section 4 establishes a relationship between the distributions
of the ranks of input and output matrices for a Gamma channel. Section 5 proves
Theorem 16, and Section 6 proves Corollary 18, giving an exact expression for the
capacity of the Gamma channels. Finally, in Section 7 we prove the results from
matrix theory that we use in earlier sections.
2 Preliminaries on finite-dimensional vector spaces
Inthissectionwediscussfinite-dimensionalvectorspacesandconsiderseveralcount-
ing problems involving subspaces and quotient spaces.
Thefollowingtwolemmasarestandardresultsinthetheoryoffinite-dimensional
vector spaces.
Lemma 1. Let V be a vector space of dimension d , and let U be a d -dimensional
V U
subspace of V. The quotient space V/U has dimension d −d .
V U
Proof. See [5, §22].
Lemma 2. Let V be a vector space and let U be a fixed subspace of V. For any
subspace W of V
dim([W])=dim(W)−dim(W ∩U),
where [W] denotes the image of W in the quotient space V/U.
Proof. This is a standard exercise.
3
The following lemma gives the number of subspaces U of V, when, given some
subspace V of V, the intersection of U and V is fixed and the image of U in the
1 1
quotient space V/V is fixed.
1
Lemma 3. Let V be a d -dimensional vector space. Let V , V be subspaces of
V 1 2
V, of dimensions d and d respectively, such that V ⊆ V . The number of d -
V1 V2 2 1 U
dimensional subspaces U ⊆ V such that U ∩V = V and the image of U in the
1 2
quotient space V/V is the fixed d −d dimensional space U′, is given by
1 U V2
q(dU−dV2)(dV1−dV2).
Proof. Fix abasisfor V ,say{b ,...,b }. Letπ :V →V/V be the mapwhich
2 1,1 1,dV2 1
takes vectors in V to their image in V/V1. For dU′ = dU −dV2, let {y1,...ydU′} be
a basis for U′, and let {b ,...,b } be some vectors in V such that π(b )=y ,
2,1 2,dU′ 2,i i
for i=1,...,dU′.
The set {b ,...,b ,b ,...,b } is a linearly independent set that forms
1,1 1,dV2 2,1 2,dU′
the basis ofa space U ofthe requiredform. Moreover,everyspace U ofthe required
formhasabasisthatcanbe constructedinthis way. The generalconstructiontakes
the basis for V2 and extends this to a basis for a space U by adding any set of dU′
vectors in V whose image under π is {y ,...,y }.
1 dU′
Given the set {b2,1,...,b2,dU′}, all other sets of dU′ vectors in V whose image
under π is {y ,...y } can be written in the form {v +b ,...,v +b } for
1 dU′ 1 2,1 dU′ 2,dU′
some v ,...,v ∈ V . Therefore, any space U of the required form has a basis
1 dU′ 1
that can be written as B = {b ,...,b ,v +b ,...,v + b } for some
1,1 1,dV2 1 2,1 dU′ 2,dU′
v ,...,v ∈V .
1 dU′ 1
Given some set v′,...,v′ ∈ V , let B′ = {b ,...,b ,v′ +b ,...,v′ +
1 dU′ 1 1,1 1,dV2 1 2,1 dU′
b2,dU′}. Now Span(B) = Span(B′) if and only if vi −vi′ ∈ V2 for i = 1,...,dU′.
That is, the sets B and B′ give rise to the same space if and only if [v ] = [v′] for
i i
i=1,...,dU′,where[v]denotesthe imageofavectorv inthequotientspaceV1/V2.
Therefore there is a bijection betweenspaces U of the requiredformand ordered
sets {[v ],...,[v ]} of elements in the quotient space V /V .
1 dU′ 1 2
For i=1,...dU′, there are qdV1−dV2 choices for [vi]∈V1/V2, thus there are
qdU′(dV1−dV2) =q(dU−dV2)(dV1−dV2). (1)
choices for the ordered set {[v ],...,[v ]}. The result follows.
1 dU′
Given a vector space V and a subspace V ⊆ V, Lemma 3 can be used to count
1
subspaces U of V when either U ∩V is fixed, or the image of U in V/V is fixed, or
1 1
when only the dimensions of these spaces are fixed. These results are given in the
following three corollaries.
Corollary 4. Let V be a d -dimensional vector space. Let V , V be subspaces
V 1 2
of V, of dimensions d and d respectively, such that V ⊆ V . The number of
V1 V2 2 1
d -dimensional subspaces U ⊆V such that U ∩V =V , is given by
U 1 2
d −d
q(dU−dV2)(dV1−dV2) V V1 .
d −d
(cid:20) U V2(cid:21)q
Proof. Consider the quotient space V/V , this is a space of dimension d −d . Let
1 V V1
U′ be a (d −d )-dimensional subspace of V/V . There are
U V2 1
d −d
V V1
d −d
(cid:20) U V2(cid:21)q
possible choices for U′, fix one such space.
ByLemma3thereareq(dU−dV2)(dV1−dV2)possibilitiesforthespaceU whoseimage
inthequotientV/V isthefixedspaceU′. Multiplyingbythenumberofpossibilities
1
for U′, yields the result.
4
Corollary 5. Let V be a d -dimensional vector space. Let V be a d -dimensional
V 1 V1
subspace of V. The number of d -dimensional subspaces U ⊆V such that the image
U
of U in the quotient space V/V1 is some fixed dU′ dimensional space U′, is given by
d
qdU′(dV1−(dU−dU′)) V1 . (2)
(cid:20)dU −dU′(cid:21)q
Proof. Let V2 be a (dU −dU′)-dimensional subspace of V1, there are
d
V1
(cid:20)dU −dU′(cid:21)q
possible choices for V , fix one.
2
By Lemma 3 there are q(dU−(dU−dU′))(dV1−(dU−dU′)) = qdU′(dV1−(dU−dU′)) possi-
bilities for the space U whose intersectionwith V is the fixed space V . Multiplying
1 2
by the number of possibilities for V , yields the result.
2
Corollary 6. Let V be a d -dimensional vector space. Let V be a d -dimensional
V 1 V1
subspace of V. For a subspace U of V, let [U] denote the image of U in the quotient
spaceV/V . Thenumberofd -dimensionalsubspacesU ⊆V suchthatdim(U∩V )=
1 U 1
d is equal to the number of d -dimensional subspaces U such that dim([U]) =
UV1 U
d −d . This number is equal to
U UV1
d −d d
q(dU−dUV1)(dV1−dUV1) V V1 V1 .
d −d d
(cid:20) U UV1(cid:21)q(cid:20) UV1(cid:21)q
Proof. Note that dim(U∩V )=d if and only if dim([U])=d −d hence the
1 UV1 U UV1
equivalence in the statement of the lemma holds. Let V be a (d )-dimensional
2 UV1
subspace of V , there are
1
d
V1
d
(cid:20) UV1(cid:21)q
possible choices for V , fix one. Next let U′ be a (d −d )-dimensional subspace
2 U UV1
of V/V . There are
1
d −d
V V1
d −d
(cid:20) U UV1(cid:21)q
possible choices for U′, fix one.
By Lemma 3 there are q(dU−dUV1)(dV1−dUV1) possibilities for the space U whose
intersection with V is the fixed space V , and image in the quotient V/V is the
1 2 1
fixed space U′. Multiplying by the number of possibilities for V and U′, yields the
2
result.
3 Matrices over finite fields
This short section describes the notation and results we use from the theory of
matrices over finite fields.
Let q be a non-trivial prime power, and let Fq be the finite field of order q. In
the introduction we defined the notation Fn×m for the set of n×m matrices with
q
entries in Fq, Fqn×m,r for the subset of Fqn×m consisting of those matrices of rank r
and GL(n,q) for the set of invertible matrices in Fn×n.
q
For a matrix M, we write rk(M) for the rank of M and we write Row(M) for
the row space of M.
The following results will be proved in Section 7.
Lemma 7. Let U be a subspace of Fm of dimension u. The number of matrices
q
M ∈ Fn×m such that Row(M) = U can be efficiently computed; it depends only on
q
q, n, m and u. We write f (u) for the number of matrices M of this form.
0
5
Lemma 8. Let U and V be subspaces of Fm of dimensions u and v respectively.
q
Let h = dim(U ∩V). Let M ∈ Fn×m be a fixed matrix such that Row(M) = U.
q
Let r be a non-negative integer. The number of matrices W ∈ Fn×m,r such that
q
Row(W +M) = V can be efficiently computed; it depends only on q, n, m, r, u, v
and h. We write f (u,v,h;r) for the number of matrices W of this form.
1
Lemma 9. Let r, r and r be non-negative integers. Let X be a fixed matrix such
W X
that rk(X)= rX. The number of matrices W ∈Fqn×m,rW such that rk(X +W)=r
can be efficiently computed; it depends only on q, n, m, r, r and r . We write
W X
f (r,r ,r ) for the number of matrices W of this form.
2 X W
By an efficient computation, we mean a polynomial (in max{n,m}) number of
arithmetic operations. In Section 7 we will prove Lemmas 7, 8 and 9 and give exact
expressions for the functions f ,f and f in terms of their inputs and the values q,
0 1 2
n and m.
4 Input and output rank distributions
A distribution PX on the input set X of the Gamma channel induces a distribution
(the input rank distribution) RX on the set of possible ranks of input matrices. Let
RY be the corresponding output rank distribution, induced from the distribution on
theoutputsetofthe Gammachannel. Akeyresult(Lemma11)isthatRY depends
on only the channel parameters and RX (rather than on PX itself). This section
aims to prove this result: it will play a vital role in the proof of Theorem 16 below.
Definition 5. Let r,r ,r ∈{0,...,min{n,m}}. Define
X B
f (r,r ,r )
2 X B
ρ(r;r ,r )= ,
X B |Fqn×m,rB|
where f2 is as defined in Lemma 9. For any fixed matrix X ∈Fqn×m,rX, ρ(r;rX,rB)
gives the proportion of matrices B ∈ Fn×m,rB with rk(X +B) = r. Let R be a
q
probability distribution on the set {0,1,...,min{n,m}} of possible ranks of n×m
matrices. Define
min{n,m}
ρ(r;r )= R(r )ρ(r;r ,r ),
X B X B
rXB=0
sothatρ(r;r )givesthe weightedaverageofthis proportionoverthepossibleranks
X
of matrices B.
Lemma10. LetX beann×mmatrixsampledfromsomedistributionPX onFqn×m.
Let B be an n×m matrix sampled from a UGR distribution with rank distribution
R, where X and B are chosen independently. Let r,r ,r ∈ {0,...,min{n,m}}.
X B
Then
ρ(r;r ,r )=Pr(rk(X+B)=r|rk(X)=r and rk(B)=r ), (3)
X B X B
and
ρ(r;r )=Pr(rk(X +B)=r|rk(X)=r ). (4)
X X
Proof. Let X be a fixed n × m matrix of rank r . Then, since B has a UGR
X
distribution,
Pr(rk(X +B)=r|rk(B)=r )
B
|{B ∈Fn×m,rB :rk(X +B)=r}|
q
=
|Fqn×m,rB|
f (r,r ,r )
2 X B
=
|Fqn×m,rB|
=ρ(r;r ,r ). (5)
X B
6
Note that (5) only depends on rk(X), not X itself. Hence
Pr(rk(X +B)=r|rk(X)=r ,rk(B)=r )
X B
= Pr(X =X)Pr(rk(X +B)=r|rk(B)=r )
B
X
X
= Pr(X =X)ρ(r;r ,r )
X B
X
X
=ρ(r;r ,r ),
X B
where the sums are over matrices X ∈Fn×m,rX. Thus (3) holds. Also
q
Pr(rk(X +B)=r|rk(X)=r )
X
min{n,m}
= R(r )ρ(r;r ,r ) (by (3))
B X B
rXB=0
=ρ(r;r ).
X
Thus (4) holds, and so the lemma follows.
Lemma 11. For the Gamma channel Γ(R) with input rank distribution RX, the
output rank distribution is given by
min{n,m}
f (r,r ,r )
2 X B
RY(r)= rXX,rB=0 RX(rX)R(rB) |Fqn×m,rB|
for r = 1,...,min{n,m}. In particular, RY depends only on the input rank distri-
bution (and the channel parameters), not on the input distribution itself.
Proof. We have that Pr(rk(X) = rX) = RX(rX) and Pr(rk(B) = rB) = R(rB).
Hence, by (3),
RY(r)=Pr(rk(Y)=r)
min{n,m}
= RX(rX)R(rB)ρ(rY;rX,rB)
rXX,rB=0
min{n,m}
f (r,r ,r )
2 X B
= rXX,rB=0 RX(rX)R(rB) |Fqn×m,rB| .
5 A UGR input distribution achieves capacity
This section shows (Theorem 16) that there exists a UGR input distribution to the
Gamma channel that achieves capacity.
Lemma 12. Let M and M′ be fixed n×m matrices of the same rank. Let B be an
n×m matrix picked from a UGR distribution, and let A be an n×n matrix picked
uniformly from GL(n,q), with B and A picked independently. Let Y =A(M +B)
and let Y′ =A(M′+B). Then
H(Y)=H(Y′).
Proof. LetAbeafixedn×ninvertiblematrix. SincethematricesAM andAM′have
the same rank, there exist invertible matrices R and C such that AM′ = RAMC.
Consider the linear transformation ϕ: Fn×m → Fn×m defined by ϕ(Y)= RYC. It
q q
is simple to check that ϕ is well defined and a bijection. Note that
ϕ(A(M +B))=RAMC+RABC
=A(M′+A−1RABC).
7
Since B is picked uniformly once its rank is determined, pre- and post-multiplying
B by fixed invertible matrices gives a uniform matrix of the same rank, therefore B
and A−1RABC have the same distribution. Now
Pr(Y =Y|A=A)
=Pr(A(M +B)=Y)
=Pr(ϕ(A(M +B))=ϕ(Y))
=Pr A(M′+A−1RABC)=ϕ(Y)
=Pr(A(M′+B)=ϕ(Y)) (6)
(cid:0) (cid:1)
=Pr(Y′ =ϕ(Y)|A=A), (7)
where(6) holdssince the distributions ofB and A−1RABC are the same. Since (7)
is true for any fixed matrix A, it follows that
Pr(Y =Y)= Pr(A=A)Pr(Y =Y|A=A)
A∈GXL(n,q)
= Pr(A=A)Pr(Y′ =ϕ(Y)|A=A)
A∈GXL(n,q)
=Pr(Y′ =ϕ(Y)). (8)
Thus Y and Y′ have the same distribution, up to relabeling by ϕ. In particular, we
find that H(Y)=H(Y′).
Definition 6. Let M be any n×m matrix of rank r. Let A be an n×n invertible
matrix chosen uniformly from GL(n,q). Let B be an n×m matrix chosen from a
UGRdistributionwithrankdistributionR,whereAandBarepickedindependently.
We define
h =H(A(M +B)).
r
Lemma 12 implies that the value h does not depend on M, only on the rank r
r
and the channel parameters q,n,m and R. The exact value of h will be calculated
r
later, in Theorem 17.
Lemma13. Consider theGammachannelΓ(R)withinputdistributionPX. Letthe
input matrix X be sampled from PX, and let Y be the corresponding output matrix.
Then
min{n,m}
H(Y|X)= RX(r)hr.
r=0
X
In particular, H(Y|X) depends only on the associated input rank distribution RX
and the channel parameters.
Proof. Choosing A and B as in the definition of the Gamma channel, we see that
H(Y|X)= P(X =X)H(A(X +B))
X∈X
X
= P(X =X)h
rk(X)
X∈X
X
min{n,m}
= RX(r)hr,
r=0
X
whichestablishesthefirstassertionofthe lemma. The secondassertionfollowssince
h depends only on r and the channel parameters.
r
8
Lemma 14. Let Y1 and Y2 be two random n×m matrices, sampled from distri-
butions with the same associated rank distribution RY. If the distribution of Y2 is
UGR then H(Y2)≥H(Y1).
Proof. For i = 1,2, we see that rk(Yi) is fully determined by Yi and so it follows
that H(Yi,rk(Yi)) = H(Yi). Therefore by the chain rule for entropy (e.g. [4, Thm.
2.2.1]),
H(Yi)=H(Yi,rk(Yi))
=H(Yi|rk(Yi))+H(rk(Yi)). (9)
Since Y2 is distributed uniformly once its rank is determined, H(Y2|rk(Y2) = r) is
maximal (e.g. [4, Thm. 2.6.4]), hence
H(Y2|rk(Y2)=r)≥H(Y1|rk(Y1)=r). (10)
Thus, using (9)
H(Y2)=H(Y2|rk(Y2))+H(rk(Y2))
min{n,m}
= (RY(r)H(Y2|rk(Y2)=r))+H(rk(Y2))
r=0
X
min{n,m}
≥ (RY(r)H(Y1|rk(Y1)=r))+H(rk(Y2)) (11)
r=0
X
=H(Y1|rk(Y1))+H(rk(Y2))
=H(Y1|rk(Y1))+H(rk(Y1)) (12)
=H(Y1),
where (11) follows from(10), and(12) follows since the rank distributions of Y1 and
Y2 are the same.
Lemma 15. Consider the Gamma channel Γ(R). If the input distribution PX is
UGR then the induced output distribution PY is also UGR.
Proof. SupposetheinputdistributionisUGR,withrankdistributionRX. We start
by showing that the distribution of X +B is UGR. Let D be any n×m matrix.
Then
Pr(X+B =D)
= Pr(X =X)Pr(X +B =D|X =X)
X∈XFqn×m
= RX(rk(X)) Pr(X +B =D),
X∈XFnq×m |Fnq×m,rk(X)|
since X is sampled from a UGR distribution. Hence
Pr(X +B =D)
min{n,m}
= RX(r) Pr(B =D−X)
Xr=0 |Fqn×m,r|X∈XFnq×m,r
min{n,m}
RX(r) R(rk(D−X))
= ,
Xr=0 |Fqn×m,r|X∈XFnq×m,r |Fqn×m,rk(D−X)|
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since X and B are independent, and since B has a UGR distribution with rank
distribution R. Now
R(rk(D−X))
X∈XFnq×m,r |Fqn×m,rk(D−X)|
min{n,m}
R(r )
= rXB=0 |{X ∈Fqn×m,r :rk(D−X)=rB}||Fqn×mB,rB|
min{n,m}
R(r )
B
= f (r ,rk(D),r)
rXB=0 2 B |Fqn×m,rB|
and so
Pr(X +B =D)
min{n,m} min{n,m}
RX(r) R(rB)
= f (r ,rk(D),r) .
Xr=0 |Fqn×m,r| rXB=0 2 B |Fqn×m,rB|
So Pr(X+B =D) does not depend onthe specific matrix D, only its rank. There-
fore, given any two n×m matrices D ,D of the same rank,
1 2
Pr(X +B =D )=Pr(X +B =D ).
1 2
Hence X +B has a UGR distribution.
Let A be a fixed n×n invertible matrix. Since X +B is picked uniformly once
its rank is determined, multiplying X +B by the invertible matrix A will give a
uniform matrix of the same rank, therefore A(X +B) has a UGR distribution. So,
defining Y = A(X +B) to be the output of the Gamma channel, we see that for
any matrix Y
Pr(Y =Y|A=A)=Pr(A(X +B)=Y)
Pr(rk(A(X +B))=rk(Y))
=
|Fnq×m,rk(Y)|
Pr(rk(Y)=rk(Y)|A=A)
= ,
|Fnq×m,rk(Y)|
where the second equality follows since A(X +B) has a UGR distribution. Thus
Pr(Y =Y)= Pr(A=A)Pr(Y =Y|A=A)
A∈GL(n,q)
X
Pr(rk(Y)=rk(Y)|A=A)
= Pr(A=A)
n×m,rk(Y)
A∈GXL(n,q) |Fq |
1
= Pr(rk(Y)=rk(Y)). (13)
|Fnq×m,rk(Y)|
Since (13) holds for all Y ∈Fn×n it follows that Y has a UGR distribution.
q
Theorem 16. For the Gamma channel Γ(R), there exists a UGR input distribu-
tion that achieves channel capacity. Moreover, given any input distribution PX with
associated rank distribution RX, if PX achieves capacity then the UGR distribution
with rank distribution RX achieves capacity.
Proof. Let X1 be a channel input, with output Y1 such that PX1 is a capacity
achieving input distribution. That is maxPX{I(X,Y)} = I(X1,Y1). Then define
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