Download On Multiplicative Matrix Channels over Finite Chain

On Multiplicative Matrix Channels
over Finite Chain Rings
Roberto W. Nóbrega∗ , Chen Feng† , Danilo Silva∗ , Bartolomeu F. Uchôa-Filho∗
∗ Department
of Electrical Engineering, Federal University of Santa Catarina, Brazil
of Electrical and Computer Engineering, University of Toronto, Canada
[email protected], [email protected], [email protected], [email protected]
† Department
Abstract—Motivated by nested-lattice-based physical-layer network coding, this paper considers communication in multiplicative matrix channels over finite chain rings. Such channels are
defined by the law Y = AX, where X and Y are the input
and output matrices, respectively, and A is called the transfer
matrix. We assume that the instances of the transfer matrix
are unknown to the transmitter, but available at the receiver. As
contributions, we obtain a closed-form expression for the channel
capacity, and we propose a coding scheme that can achieve this
capacity with polynomial time complexity. Our results extend the
corresponding ones for finite fields.
Index Terms—Finite chain rings, multiplicative matrix channels, random linear network coding.
I. I NTRODUCTION
A multiplicative matrix channel (MMC) over a finite
field Fq is a communication channel in which the input
are related by
and the output Y ∈ Fm×`
X ∈ Fn×`
q
q
Y = AX,
(1)
is called the transfer matrix. Such channels
where A ∈ Fm×n
q
turn out to be suitable models for the end-to-end communication channel between a source node and a sink node in an
error-free, erasure-prone network performing random linear
network coding [1]. In this context, X is the matrix whose
rows are the n packets (of length `) transmitted by the source
node, Y is the matrix whose rows are the m packets received
by the sink node, and A is a random matrix whose entries
are determined by factors such as the network topology and
the random choices of the network coding coefficients. Note
that each packet can be viewed as an element of the message
space Ω = F`q , a finite vector space.
The present work considers MMCs over finite chain rings
(of which finite fields are a special case). The motivation
comes from physical-layer network coding [2] employing
compute-and-forward [3]. Indeed, [4] shows that in a wireless network employing compute-and-forward over a generic
nested lattice, the end-to-end communication channel between
a source node and a sink node can still be modeled1 by the
The work of R. W. Nóbrega, D. Silva, and B. F. Uchôa-Filho was partly
supported by CNPq-Brazil.
1 Note that any additive error introduced at the physical layer may be
avoided, at each relay node, by employing a linear error-detecting code over
the underlying ring.
same channel law (1). In this case, though, the underlying ring
is a not a finite field, but a principal ideal domain T (typically
the integers Z, the Gaussian integers Z[i], or the Eisenstein integers Z[ω]), with the corresponding message space Ω being a
finite T -module. As such, Ω ∼
= T /hd1 i×T /hd2 i×· · ·×T /hd` i,
where d1 , d2 , . . . , d` ∈ T are non-zero non-unit elements
satisfying d1 | d2 | · · · | d` . A special situation commonly
found in practice is when the di s are all powers of a given
prime of T . In this case, the message space Ω can also be
viewed as a finite R-module, where R = T /hd` i is a finite
chain ring.
Finite-field MMCs have been studied under a probabilistic
approach according to different assumptions on the transfer
matrix [5]–[10]. In this work, we consider finite-chain-ring
MMCs with channel side information at the receiver (CSIR),
that is, we assume that the instances of the transfer matrix A
are unknown to the transmitter, but available at the receiver.
Besides that, we impose no restrictions on the statistics of A,
except that it must be independent of X. As contributions,
we obtain a closed-form expression for the channel capacity
(§V-A), and we propose a coding scheme for the channel,
which combines several codes over a finite field to obtain
a code over a finite chain ring (§V-B and §V-C). We then
show that the scheme can achieve the channel capacity with
polynomial time complexity, and that it does not necessarily
require the complete knowledge of the probability distribution
of A; only the expected value of its rank (or, rather, its
“shape”—see Section II) is needed (§V-D). The results are
then naturally adapted to the non-coherent scenario, i.e., the
case in which the instances of the transfer matrix are unknown
to both the transmitter and receiver (§V-E). Our results extend
(and make use of) some of those obtained by Yang et al.
in [7], which addresses the finite field case. It is also worth
mentioning that a generalization of the results in [6] from finite
fields to finite chain rings is presented in [11].
The remainder of this paper is organized as follows. Section II reviews basic concepts on finite chain rings and linear
algebra over them. Section III presents the channel model.
Section IV reviews some of the existing results on MMCs over
finite fields, and Section V contains our main contributions
about MMCs over finite chain rings. Finally, Section VI
concludes the paper.
II. BACKGROUND ON F INITE C HAIN R INGS
We now present some basic results on finite chain rings and
linear algebra over them. For more details, we refer the reader
to [12]–[15]. By the term ring we always mean a commutative
ring with identity 1 6= 0.
A. Finite Chain Rings
A ring R is called a chain ring if, for any two ideals I, J
of R, either I ⊆ J or J ⊆ I. It is known that a finite ring R
is a chain ring if and only if R is both principal (i.e., all of its
ideals are generated by a single element) and local (i.e., the
ring has a unique maximal ideal). Let π ∈ R be any generator
for the maximal ideal of R, and let s be the nilpotency index
of π (i.e., the smallest integer s such that π s = 0). Then, one
can show that R has precisely s + 1 ideals, namely,
R = hπ 0 i ⊃ hπ 1 i ⊃ · · · ⊃ hπ s−1 i ⊃ hπ s i = {0},
where hxi denotes the ideal generated by x ∈ R. We call
the invariant s the depth of the chain ring R. Furthermore, it
is also known that the quotient R/hπi is a field, called the
residue field of R. If q = |R/hπi|, then the size of each ideal
of R is |hπ i i| = q s−i , 0 ≤ i ≤ s; in particular, |R| = q s . Note
that, if s = 1, then π = 0 and R is a finite field.
An example of a chain ring is Z8 = {0, 1, . . . , 7}, the ring of
integers modulo 8. Its ideals are h1i = Z8 , h2i = {0, 2, 4, 6},
h4i = {0, 4}, and h0i = {0} (so that s = 3), and its residue
field is Z8 /h2i ∼
= F2 (so that q = 2).
For the remainder of this paper, R denotes a chain ring with
depth s and residue field of size q. Also, π ∈ R denotes a fixed
generator for its maximal ideal, and Γ ⊆ R denotes a fixed
set of coset representatives for the residue field of R. Without
loss of generality, assume 0 ∈ Γ.2
Lemma 1. Every element x ∈ R can be written uniquely as
x=
s−1
X
x(i) π i ,
i=0
where x
(i)
∈ Γ, for 0 ≤ i < s.
We next mention a few basic results that help us compute
with π-adic expansions. The proof is omitted due to lack of
space.
Lemma 2. Let x, y, z ∈ R. Then,
(i+j)
1) xπ i
= x(j) , for 0 ≤ j < s − i; and
i
2) (x + yπ + zπ i+1 )(i) ≡π x(i) + y (0) .
B. Modules over Finite Chain Rings
By an s-shape we mean a non-decreasing sequence of s
non-negative integers. Let µ = (µ0 , µ1 , . . . , µs−1 ) be an sshape. We define
Rµ , h1i × · · · × h1i × hπi × · · · × hπi × · · ·
|
{z
} |
{z
}
µ1 −µ0
µ0
× hπ
|
s−1
i × · · · × hπ s−1 i .
{z
}
µs−1 −µs−2
Clearly, being a Cartesian product of ideals, Rµ is a finite Rmodule. Conversely, every finite R-module M is isomorphic
to Rµ for some unique s-shape µ. We call µ the shape of M ,
and write µ = shape M . We have
|Rµ | = q |µ| ,
(2)
where |µ| is defined as |µ| = µ0 + µ1 + · · · + µs−1 . For
convenience, we set µ−1 = 0.
Let µ be an s-shape. We define µ − n = (µ0 − n, µ1 −
n, . . . , µs−1 − n), which is also an s-shape. For convenience,
we may write the s-shape (m, m, . . . , m) simply as m. According to this convention, Rm stands for the same object,
whether m is interpreted as an integer or as an s-shape.
C. Matrices over Finite Chain Rings
For any subset S ⊆ R, we denote by S m×n the collection
of all m×n matrices with entries in S. The set of all invertible
n × n matrices over R is called the general linear group of
degree n over R, and is denoted by GLn (R).
Let A ∈ Rm×n , and set r = min{n, m}. A diagonal matrix
(not necessarily square)
D = diag(d1 , d2 , . . . , dr ) ∈ Rm×n
for 0 ≤ i ≤ s. One can show that xi ≡πi x for all x ∈ R,
where ≡a denotes congruence modulo a (i.e., x ≡a y if and
only if x − y ∈ hai). In particular, x(0) = x1 ≡π x.
is called a Smith normal form of A if there exist matrices
P ∈ GLm (R) and Q ∈ GLn (R) (not necessarily unique)
such that A = P DQ and d1 | d2 | · · · | dr . It is known
that matrices over principal rings (in particular, finite chain
rings) always have a Smith normal form, which is unique up
to multiplication of the diagonal entries by units. In this work,
we shall require such entries be powers of π ∈ R; by doing
so, the Smith normal form becomes (absolutely) unique.
Let row A and col A denote the row and column span of
A ∈ Rm×n , respectively. By using the Smith normal form, we
can easily prove that row A is isomorphic to col A. We define
the shape of A as shape A = shape(row A) = shape(col A).
Moreover, µ = shape A if and only if the Smith normal form
of A is given by
particularly nice, canonical choice for Γ is Γ(R) = {x ∈ R : xq = x},
known as the Teichmüller coordinate set of R.
diag(1, . . . , 1, π, . . . , π , . . . , π s−1 , . . . , π s−1 , 0, . . . , 0),
| {z } | {z }
|
{z
} | {z }
The above expression is known as the π-adic expansion
of x (with respect to Γ). For example, the 2-adic expansion of
6 ∈ Z8 with respect to Γ = {0, 1} is 6 = 0 · 20 + 1 · 21 + 1 · 22 ,
i.e., the standard binary expansion of 6.
Note that the uniqueness of the π-adic expansion (given Γ)
allows us to define the maps (·)(i) : R → Γ, for 0 ≤ i < s.
We also define
i−1
X
x(j) π j ,
xi =
j=0
2A
µ0
µ1 −µ0
µs−1 −µs−2
r−µs−1
where r = min{n, m}. For example, consider the matrix
4 3 6
A=
6 7 2
over Z8 . Then, A = P DQ, where
1
P =
1
0
,
1
1
D=
0
0
2
0
,
0
The rate of the code C is defined by R(C) = (log |C|)/N ,
and its probability of error, denoted by Pe (C), is defined as
usual [17]. When N = 1, we say that C is a one-shot code;
otherwise, we say that C is a multi-shot code.
IV. R EVIEW OF THE MMC OVER A F INITE F IELD

4
Q = 1
5
3
2
6

6
6 ,
3
so that shape A = (1, 2, 2). We also define the kernel of A
as usual, that is, ker A = {x ∈ Rn : Ax = 0}. From the first
isomorphism theorem [16, §10.2], col A ∼
= Rn / ker A.
Let λ be an s-shape. We denote by Rn×λ the subset of
matrices in Rn×` whose rows are elements of Rλ , where ` =
λs−1 . Clearly, |Rn×λ | = q n|λ| . Finally, we extend the π-adic
expansion map (·)(i) to matrices over R in an element-wise
fashion. Thus, A ∈ Rn×λ if and only if A(i) = Bi 0 ∈
Γn×` , for some Bi ∈ Γn×λi , 0 ≤ i < s.
III. C HANNEL M ODEL
Throughout this paper, bold symbols are used to represent
random entities, while regular symbols are used for their
samples.
Recall that a discrete memoryless channel (DMC) [17] with
input x and output y is defined by a triplet (X , py|x , Y),
where X and Y are the channel input and output alphabets,
respectively, and py|x , called the channel transition probability, gives the probability that y = y ∈ Y is received given that
x = x ∈ X is sent. The channel is memoryless in the sense
that the output symbol at a given time depends only on the
input at that time and is conditionally independent of previous
inputs or outputs. The capacity of the DMC is given by
C = max I(x; y),
px
where I(x; y) is the mutual information between x and y, and
the maximization is over all possible input distributions px .
Let R be a finite chain ring, let n and m be positive
integers, and let λ be an s-shape. Also, let pA be a probability
distribution over Rm×n . From these, we can define the MMC
with CSIR over R as a DMC with input X ∈ Rn×λ , output
(Y , A) ∈ Rm×λ × Rm×n , and channel transition probability
(
pA (A), if Y = AX,
pY ,A|X (Y, A|X) =
0,
otherwise.
In this work, we shall denote the channel just defined simply
by MMCCSIR (A, λ). We also make use of the random variable
ρ = shape A, distributed according to
X
pρ (ρ) =
pA (A),
A: shape A=ρ
Finally, set ` = λs−1 (interpreted as the packet length).
An matrix (block) code of length N is defined by a pair
(C, Φ), where C ⊆ (Rn×λ )N is called the codebook, and Φ :
(Rm×λ × Rm×n )N → C is called the decoding function. We
sometimes abuse the notation and write C instead of (C, Φ).
In this section, we briefly review some of the existing results
about the MMC with CSIR over a finite field (i.e., R = Fq ).
Note that, in this case, s = 1, λ = `, and ρ = rank A , r.
The following result is due to Yang et al. [7], [8].
Theorem 3. [7, Prop. 1] The capacity of MMCCSIR (A, `) is
given by
C = E[r]`,
in q-ary digits per channel use, and is achieved if the input
is uniformly distributed over Fn×`
. In particular, the capacity
q
depends on pA only through E[r].
Also in [7], [8], two multi-shot coding schemes for MMCs
over finite fields are proposed, which are able to achieve
the channel capacity given in Theorem 3. The first scheme
makes use of rank-metric codes [18] and requires ` ≥ n in
order to be capacity-achieving; the second scheme is based on
random coding and imposes no restriction on `. Both schemes
have polynomial time complexity. Also important, both coding
schemes are “universal” in the sense that only the value of
E[r] is taken into account in the code construction (the full
knowledge of pA , or even pr , is not required).
V. T HE MMC OVER A F INITE C HAIN R ING
Consider again the case of a general finite chain ring R.
A. Channel Capacity
The following result generalizes Theorem 3.
Theorem 4. The capacity of MMCCSIR (A, λ) is given by
C=
s−1
X
E[ρs−i−1 ]λi ,
i=0
in q-ary digits per channel use, and is achieved if the input is
uniformly distributed over Rn×λ . In particular, the capacity
depends on pA only through E[ρ].
In order to prove this theorem, we need the following
lemma.
Lemma 5. Let X ∈ Rn×λ be a random matrix, let A ∈
Rm×n be any fixed matrix, and let ρ = shape A. Define Y =
AX ∈ Rm×λ . Then,
H(Y ) ≤
s−1
X
ρs−i−1 λi ,
i=0
where equality holds if X is uniformly distributed over Rn×λ .
Proof. Note that X and Y can be expressed as
X = X0 X1 · · · Xs−1 ,
Y = Y0 Y1 · · · Ys−1 ,
where Xi ∈ hπ i in×(λi −λi−1 ) and Yi ∈ hπ i im×(λi −λi−1 ) , for
0 ≤ i < s. We have
Yi = AXi ,
so that the support of each of the columns of Yi is a subset
of col π i A. We have shape π i A = (0, . . . , 0, ρ0 , . . . , ρs−i−1 ),
so that, from (2), we have | col π i A| = q ρ0 +···+ρs−i−1 . Therefore, the support of Y has size at most
s−1
Y
i
| col π A|
λi −λi−1
i=0
=
s−1
Y
q
(ρ0 +···+ρs−i−1 )(λi −λi−1 )
i=0
Ps−1
=q
i=0
Lemma 6. Let y ∈ Rn and A ∈ GLn (R). Let x ∈ Rn be the
(unique) solution of Ax = y. Then, the π-adic expansion of x
can be obtained recursively from
(i)
A(0) x(i) ≡π y (i) − Axi ,
for 0 ≤ i < s.
Proof. For 0 ≤ i < s, we have
y = Ax = A
i−1
X
x(j) π j + Ax(i) π i + A
j=0
ρs−i−1 λi
,
from which the inequality follows.
Now suppose X is uniformly distributed over Rn×λ . This
means that Xi is uniformly distributed over hπ i in×(λi −λi−1 ) .
One may show that there exists Xi0 uniformly distributed over
Rn×(λi −λi−1 ) such that Xi = π i Xi0 . Let y denote a column
of Yi , whose support is col π i A. Since Yi = AXi = π i AXi0 ,
we have, for every y ∈ col π i A,
|{x0 ∈ Rn : π i Ax0 = y}|
|Rn |
| ker π i A|
=
|Rn |
1
=
,
| col π i A|
Pr[y = y] =
that is, y is uniformly distributed over its support. Therefore,
Y itself is also uniformly distributed over its support. This
concludes the proof.
We can now prove Theorem 4.
j=i+1
y (i) ≡π Axi
(i)
+ Ax(i)
(0)
.
After simplifying and rearranging we get the equation displayed on the lemma. Since A(0) ∈ GLn (Fq /hπi), we can
compute, recursively, x(0) , x(1) , . . . , x(s−1) .
The second problem deals with the solution of diagonal
systems of linear equations. Let Mj:j 0 denote the sub-matrix
of M consisting of rows j up to, but not including, j 0 , where
we index the matrix entries starting from 0.
Lemma 7. Let Y ∈ Rm×λ and D ∈ Rm×n , where D is the
Smith normal form of itself and has shape ρ. If Y = DX,
then


(i)
Y0:ρ0


(i+1)


Yρ0 :ρ1
(i)
,
X0:ρs−i−1 = 
..




.
(i+s−1)
Yρs−i−2 :ρs−i−1
for 0 ≤ i < s.
Proof. Note that Y = DX is equivalent to
Y0:ρ0 = X0:ρ0 ,
Yρ0 :ρ1 = πXρ0 :ρ1 ,
..
.
I(X; Y , A) = I(X; Y |A) + I(X; A)
= H(Y |A),
x(j) π j ,
so that, from Lemma 2,
Proof of Theorem 4. The channel mutual information is given
by
= H(Y |A) − H(Y |X, A) + I(X; A)
s−1
X
Yρs−2 :ρs−1 = π s−1 Xρs−2 :ρs−1 .
From Lemma 2, this implies
where H(Y |X, A) = 0 since Y = AX, and I(X; A) = 0
since X and A are independent. Thus,
X
I(X; Y , A) = H(Y |A) =
pA (A)H(Y |A = A),
A
and the result follows from Lemma 5.
B. Auxiliary Results
Before we describe the proposed coding scheme, we first
present two simple lemmas regarding the solution of systems
of linear equations over a finite chain ring. These results will
serve as a basis for the coding scheme. The first problem turns
a system of linear equations over the chain ring into multiple
systems over the residue field.
(i)
(i)
X0:ρ0 = Y0:ρ0 ,
Xρ(i)
0 :ρ1
=
..
.
Yρ(i+1)
,
0 :ρ1
0 ≤ i < s,
0 ≤ i < s − 1,
..
.
Xρ(i)
= Yρ(i+s−1)
, 0 ≤ i < 1,
s−2 :ρs−1
s−2 :ρs−1
from which the result follows.
Example: Let R = Z8 , with π = 2 and Γ = {0, 1}. Let
n = 5, m = 4, and λ = (3, 4, 6). Suppose ρ = (1, 3, 4), so
that D = diag(1, 2, 2, 4) ∈ Z4×5
. Also, suppose
8


6 7 1 2 0 4
6 4 2 0 0 0

Y =
0 2 6 4 0 0 .
4 0 4 0 0 0
From this we can conclude

0
1

X (0) = 
0
1
∗

1
1

X (1) = 
0
∗
∗

1
∗

X (2) = 
∗
∗
∗
that
1
0
1
0
∗
1
1
1
1
∗
1
1
0
∗
∗
0
0
1
∗
∗
1
0
0
∗
∗
1
∗
∗
∗
∗
0
∗
∗
∗
∗
0
∗
∗
∗
∗
Step 2. Let ρ = shape A = shape D. Define X̃ , QX ∈
Rn×λ (which is unknown to the receiver) and Ỹ , P −1 Y ∈
Rm×λ (which is calculated at the receiver), so that Y = AX
is equivalent to
Ỹ = DX̃.



,


From this equation, the decoder can obtain partial information
(i)
about X̃. More precisely, it can compute X̃ρs−i−1 ×λi , for 0 ≤
i < s, according to Lemma 7.



,


0
∗
∗
∗
∗
(i)
Step 3. In possession of X̃ρs−i−1 ×λi , for 0 ≤ i < s, the
decoder will then try to decode X based on the equation
X̃ = QX,

1
∗

∗
,
∗
∗
in a multistage fashion. Indeed, similarly to Lemma 6, we
have, for 0 ≤ i < s,
(i)
X̃ (i) − QX i
≡π Q(0) X (i) .
where ∗ denotes unknown entries and blank spaces denote zero
entries. Note that the unknown entries are due to ρ = shape D,
while blank entries are due to λ (see §II-C).
C. Coding Scheme
We now propose a coding scheme for the channel, which is
based on the π-adic expansion discussed in §II-A and in the
ideas of the previous subsection. For simplicity of exposition,
we will start by describing the scheme for the particular case of
one-shot codes. The general case will be discussed afterwards.
From now on, let Fq = R/hπi.
Let C0 , C1 , . . . , Cs−1 (referred to as component codes) be a
sequence of one-shot matrix codes over the residue field Fq ,
where each Ci , for 0 ≤ i < s, is a code for MMCCSIR (Ai , λi ),
for some transfer matrix Ai ∈ Fm×n
. We will combine these
q
component codes to obtain a matrix code C over the chain
ring R for MMCCSIR (A, λ). We refer to C as the composite
code.
1) Codebook: Denote by ϕ : R → Fq the natural projection
map from R onto Fq . Also, denote by ϕ̄ : Fq → Γ the coset
representative selector map, with the property that ϕ(ϕ̄(x)) =
x for all x ∈ Fq . The codebook C ⊆ Rn×λ is defined by
(s−1
)
X
(i) i
C=
X π : Xi ∈ Ci , 0 ≤ i < s ,
i=0
where
X (i) = ϕ̄(Xi )
0 ∈ Γn×` .
(3)
It should be clear that the codewords in C indeed satisfy the
constraints of Rn×λ .
2) Decoding: We now describe the decoding procedure.
Intuitively, the decoder decomposes a single MMC over the
chain ring into multiple MMCs over the residue field. In the
following, Mj×k denotes the upper-left j ×k sub-matrix of M .
Step 1. The decoder, which knows the transfer matrix A,
starts by computing its Smith normal form D ∈ Rm×n . It
also computes P ∈ GLm (R) and Q ∈ GLn (R) such that
A = P DQ.
Considering only the ρs−i−1 topmost rows (since the remaining rows are unknown), and keeping only the λi leftmost
columns (since the remaining columns are already known to
be zero), we get
(i)
(i)
(0)
(i)
i
≡π Qρs−i−1 ×n Xn×λi .
X̃ρs−i−1 ×λi − Qρs−i−1 ×n Xn×λi
Finally, projecting into Fq (that is, applying ϕ to both sides),
and appending enough zero rows (in order to obtain an m × n
system) gives
Yi = Ai Xi ,
(4)
i
and Ai ∈ Fqm×n are defined by
where Yi ∈ Fm×λ
q
"
(i) #
(i)
i
ϕ X̃ρs−i−1 ×λi − ϕ Qρs−i−1 ×n Xn×λi
Yi =
, (5)
0
and
Ai =
ϕ Qρs−i−1 ×n
0
.
(6)
Note that Yi can only be calculated after X0 , X1 , . . . , Xi−1
are known. Therefore, in this step the decoder obtains, successively, estimates of X0 , X1 , . . . , Xs−1 from (4). Finally, it
computes an estimate of X according to (3) and the π-adic
expansion.
3) Extension to the Multi-Shot Case: We finally consider
the multi-shot case. Let C0 , C1 , . . . , Cs−1 be a sequence of
N -shot matrix codes (the component codes), where Ci ⊆
i N
(Fn×λ
) , for 0 ≤ i < s. The codewords of the composq
ite code C are then given by (X(1), X(2), . . . , X(N )) ∈
(Rn×λ )N , where X(j) is obtained from the j-th coordinates
of the codewords of the component codes, similarly to the
one-shot case.
Proceeding similarly to Steps 1 and 2 above, the de(i)
coder obtains X̃ρs−i−1 ×λi (j), for 0 ≤ i < s and
j = 1, . . . , N , and Q(j), for j = 1, . . . , N . Step 3 is
also similar, with the important detail that the whole sequence (Xi (1), Xi (2), . . . , Xi (N )) ∈ Ci is decoded from
(Yi (1), Yi (2), . . . , Yi (N )) and (Ai (1), Ai (2), . . . , Ai (N )) by
using the decoder of Ci , before proceeding to stage i + 1.
D. Universality, Rate, Probability of Error, and Complexity
From the proposed coding scheme, it is now clear that the
i-th component code should be aimed at MMCCSIR (Ai , λi ),
where Ai ∈ Fm×n
is defined in (6). In principle, we could
q
calculate the probability distribution of Ai provided we have
access to the probability distribution of A. Nevertheless, if
we employ one of the coding schemes proposed in [7] (see
Section IV), then the particular probability distribution of Ai
becomes unimportant once we know the expected value of its
rank. From (6), we have rank Ai = ρs−i−1 , so that, in this
case, only the knowledge of E[ρ] is needed. Thus, the proposed coding scheme is “universal,” provided the component
codes are also universal.
Let C denote the composite code obtained from the component codes C0 , C1 , . . . , Cs−1 . Then, the C has a rate of
R(C) = R(C0 ) + R(C1 ) + · · · + R(Cs−1 ),
in q-ary digits. Also, from the union bound, the probability of
error is upper-bounded by
Pe (C) ≤ Pe (C0 ) + Pe (C1 ) + · · · + Pe (Cs−1 ).
Thus, if each Ci is capacity-achieving in MMCCSIR (Ai , λi ),
we have R(Ci ) arbitrarily close to E[ρs−i−1 ]λi and Pe (Ci )
arbitrarily close toPzero, for 0 ≤ i < s. Therefore, R(C) is
arbitrarily close to i E[ρs−i−1 ]λi (the channel capacity), and
Pe (C) is arbitrarily close to zero.
The computational complexity of the scheme is simply the
sum of the individual computational complexities of each
component code, plus the cost of calculating the Smith normal
form of A (which can be done with O(nm min{n, m}) operations in R), the cost of calculating Ỹ (taking O(m2 (m + `))
operations), and the cost of s − 1 matrix multiplications and
additions in (5) (taking O(n2 `) operations each).
E. Extension to the Non-Coherent Scenario
So far we have only considered the case where channel side
information is available at the receiver. Nevertheless, as mentioned in the introduction, we can still reuse the coding scheme
proposed in §V even in a non-coherent scenario, by means of
channel sounding (also known as channel training). In this
technique, the instances of A are provided to the receiver by
introducing headers in the transmitted
matrix X ∈ Rn×λ , that
0
is, by setting X = I X , where I ∈ Rn×n is the identity
matrix, and X 0 ∈ Rn×(λ−n) is a payload matrix coming from
a matrix code. For this to work, we clearly need λ0 ≥ n. Note
that channel sounding introduces an overhead of n2 symbols.
Nevertheless, the overhead can be made negligible if we are
allowed to arbitrarily increase the packet length, that is, the
proposed scheme can be capacity-achieving in this asymptotic
scenario.
VI. C ONCLUSIONS
In this work we investigated multiplicative matrix channels
over finite chain rings, which have practical applications in
nested-lattice-based physical-layer network coding. As contributions, the channel capacity has been determined, which generalizes the corresponding result for finite fields. Furthermore,
a capacity-achieving coding scheme was proposed, combining
several codes over the residue field to obtain a new code over
the chain ring.
Several points are still open. The capacity of the noncoherent MMC, a problem addressed in [7], [9] for the case
of finite fields, still needs to be generalized for the case of
finite chain rings. In this line, the “uniform given shape”
distribution seems to be of fundamental importance. Also,
designing capacity-achieving coding schemes for the noncoherent MMC with small λ is still an open problem, even
in the finite-field case.
R EFERENCES
[1] R. Koetter and F. R. Kschischang, “Coding for errors and erasures in
random network coding,” IEEE Transactions on Information Theory,
vol. 54, pp. 3579–3591, Aug. 2008.
[2] S. C. Liew, S. Zhang, and L. Lu, “Physical-layer network coding:
Tutorial, survey, and beyond,” Physical Communication, vol. 6, pp. 4–
42, May 2013.
[3] B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interference through structured codes,” IEEE Transactions on Information
Theory, vol. 57, pp. 6463–6486, Oct. 2011.
[4] C. Feng, D. Silva, and F. R. Kschischang, “An algebraic approach to
physical-layer network coding,” To appear in the IEEE Transactions on
Information Theory.
[5] M. Jafari Siavoshani, S. Mohajer, C. Fragouli, and S. Diggavi, “On
the capacity of non-coherent network coding,” IEEE Transactions on
Information Theory, vol. 57, pp. 1046–1066, Feb. 2011.
[6] D. Silva, F. R. Kschischang, and R. Koetter, “Communication over
finite-field matrix channels,” IEEE Transactions on Information Theory,
vol. 56, pp. 1296–1305, Mar. 2010.
[7] S. Yang, S.-W. Ho, J. Meng, E.-h. Yang, and R. W. Yeung, “Linear
operator channels over finite fields,” Computing Research Repository
(CoRR), vol. abs/1002.2293, Apr. 2010.
[8] S. Yang, J. Meng, and E.-h. Yang, “Coding for linear operator channels
over finite fields,” in Proceedings of the 2010 IEEE International
Symposium on Information Theory (ISIT’10), (Austin, Texas), pp. 2413–
2417, June 2010.
[9] R. W. Nóbrega, D. Silva, and B. F. Uchôa-Filho, “On the capacity of
multiplicative finite-field matrix channels,” Computing Research Repository (CoRR), vol. abs/1105.6115, Apr. 2013. To appear in the IEEE
Transactions on Information Theory.
[10] S. Yang, S.-W. Ho, J. Meng, and E.-h. Yang, “Capacity analysis of linear
operator channels over finite fields,” Computing Research Repository
(CoRR), vol. abs/1108.4257, Dec. 2012.
[11] C. Feng, R. W. Nóbrega, F. R. Kschischang, and D. Silva, “Communication over finite-chain-ring matrix channels,” Computing Research
Repository (CoRR), vol. abs/1304.2523, Apr. 2013. Submitted to the
IEEE Transactions on Information Theory.
[12] B. R. McDonald, Finite Rings with Identity, vol. 28 of Monographs and
Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., 1974.
[13] A. A. Nechaev, “Finite rings with applications,” in Handbook of Algebra
(M. Hazewinkel, ed.), vol. 5, pp. 213–320, North-Holland, 2008.
[14] T. Honold and I. Landjev, “Linear codes over finite chain rings,” The
Electronic Journal of Combinatorics, vol. 7, 2000.
[15] W. C. Brown, Matrices over Commutative Rings, vol. 169 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker,
Inc., 1992.
[16] D. S. Dummit and R. M. Foote, Abstract Algebra. John Wiley and Sons,
3rd ed., 2004.
[17] T. M. Cover and J. A. Thomas, Elements of Information Theory. WileyInterscience, 2nd ed., 2006.
[18] D. Silva, F. R. Kschischang, and R. Koetter, “A rank-metric approach
to error control in random network coding,” IEEE Transactions on
Information Theory, vol. 54, pp. 3951–3967, Sept. 2008.