A Deliberate Bit Flipping Coding Scheme for
Data-Dependent Two-Dimensional Channels
Item Type Article
Authors Bahrami, Mohsen; Vasic, Bane
Citation Bahrami, M., & Vasić, B. (2019). A deliberate bit flipping coding
scheme for data-dependent two-dimensional channels. IEEE
Transactions on Communications.
DOI 10.1109/tcomm.2019.2957086
Publisher Institute of Electrical and Electronics Engineers (IEEE)
Journal IEEE TRANSACTIONS ON COMMUNICATIONS
Rights © 2019 IEEE.
Download date 10/08/2022 02:32:19
Item License http://rightsstatements.org/vocab/InC/1.0/
Version Final accepted manuscript
Link to Item http://hdl.handle.net/10150/641490
1
A Deliberate Bit Flipping Coding Scheme for
Data-Dependent Two-Dimensional Channels
Mohsen Bahrami, Student Member, IEEE, and Bane Vasi
´
c, Fellow, IEEE
Abstract—In this paper, we present a deliberate bit flipping
(DBF) coding scheme for binary two-dimensional (2-D) channels,
where specific patterns in channel inputs are the significant cause
of errors. The idea is to eliminate a constrained encoder and,
instead, embed a constraint into an error correction codeword
that is arranged into a 2-D array by deliberately flipping the
bits that violate the constraint. The DBF method relies on the
error correction capability of the code being used so that it
should be able to correct both deliberate errors and channel
errors. Therefore, it is crucial to flip minimum number of bits
in order not to overburden the error correction decoder. We
devise a constrained combinatorial formulation for minimizing
the number of flipped bits for a given set of harmful patterns.
The generalized belief propagation algorithm is used to find
an approximate solution for the problem. We evaluate the
performance gain of our proposed approach on a data-dependent
2-D channel, where 2-D isolated-bits patterns are the harmful
patterns for the channel. Furthermore, the performance of the
DBF method is compared with classical 2-D constrained coding
schemes for the 2-D no isolated-bits constraint on a memoryless
binary symmetric channel.
Index Terms—Data dependent channels, constrained coding,
probabilistic inference, graphical models, and generalized belief
propagation (GBP).
I. INTRODUCTION
Recent advances in magnetic recording systems [3], [4],
optical recording devices [5] and flash memory drives [6]
necessitate to study two-dimensional (2-D) coding techniques
for reliable storage/retrieval of user data. Most channels in
such systems introduce errors in messages in response to
certain data patterns, and messages containing these patterns
are more prone to errors than others. For example, in a
single-level cell flash memory channel, inter-cell interference
(ICI) is at its maximum when 101 patterns are programmed
over adjacent cells in either horizontal or vertical directions
[7]–[9]. As another example, in two-dimensional magnetic
recording channels, 2-D isolated-bits patterns [10] are shown
empirically to be the dominant error event, and during the
read-back process inter-symbol interference (ISI) and inter-
track interference (ITI) arise when these patterns are recorded
over the magnetic medium. Shannon in his seminal work [11]
presented two techniques for reliable transmission of messages
over noisy channels, namely error correction coding and con-
strained coding. In the first method, messages are protected via
an error correction code (ECC) from random errors which are
M. Bahrami and B. Vasi
´
c are with the Department of Electrical and
Computer Engineering, University of Arizona, Tucson, AZ, 85721 USA (e-
mails: bahrami@email.arizona.edu and vasic@ece.arizona.edu). This work is
supported by the National Science Foundation under grants ECCS-1500170
and SaTC-1813401. Parts of this paper were presented in IEEE GLOBECOM
conference 2016 [1] and 8th International Conference on Algebraic Informat-
ics (CAI 2019) [2].
independent of input data. The theory of ECCs is well studied,
and efficient code construction methods are developed for sim-
ple binary channels, additive white Gaussian noise (AWGN)
channels and partial response channels [12], etc. On the other
hand, constrained coding reduces the likelihood of corruption
by removing problematic patterns before transmission over
data-dependent channels. Prominent examples of constraints
include a family of binary one-dimensional (1-D) and 2-D
(d, k)-run-length-limited (RLL) constraints [13], [14] which
improves resilience to ISI timing recovery and synchronization
for bandwidth limited partial response channels, where d and
k represent the minimum and maximum number of admissible
zeros between two successive ones in any direction of array.
In principle, the ultimate coding approach for such data-
dependent channels is to design a set of sufficiently distinct
error correction codewords that also satisfy channel constraints
[15], [16]. Designing channel codewords satisfying both ECC
and channel constraints is important as it would achieve the
channel capacity [17]. However, in practice this is difficult, and
we rely on sub-optimal methods such as forward concatenation
method (standard concatenation) [18], reverse concatenation
method (modified concatenation) [19], [20], and combinations
of these approaches [21], [22].
As discussed earlier, constrained codes have been used
to overcome effects of harmful patterns in 1-D information
storage systems. In [23], a systematic approach for designing
1-D constrained codes known as the state splitting algorithm
is established. Marcus et al. used the results of the state
splitting algorithm to design an encoder in the form of
a finite state machine and a sliding window decoder with
limited error propagation [24]. The theory of 1-D constrained
coding is mature as well as practical aspects of 1-D code
and decoder design. However, for the 2-D case it remains
a challenge to design efficient, fixed-rate encoding and de-
coding algorithms (due to difficulty of certain problems that
link to 2-D constraints compared to to the 1-D case [25],
[26]). A number of variable-rate encoding methods have
been proposed for 2-D constrained channels, including bit-
stuffing encoders [10], [27]–[29] and tiling based encoders
[30], [31]. Furthermore, various row-by-row coding methods
for specific 2-D constraints were presented in [32], [33]. Vasi
´
c
and Pedagani proposed an alternative approach in [34], known
as deliberate bit flipping (DBF), for applying binary 1-D
(0, k)-RLL constraint to error correction codewords (when k
is large e.g., k = 15) to overcome the non-linear effects of 1-
D constrained codes. Using a (0, k)-RLL constraint monitor,
a deliberate bit error is introduced into an error correction
codeword whenever the number of consecutive zeros in the
codeword reaches k. The method only relies on the capability
2
of the ECC to correct both the deliberate errors and channel
errors at the receiver. In [35]–[37], the problem of number
of deliberate bit errors for imposing (0, k)-RLL constraint
into low-density parity-check (LDPC) codewords was partially
addressed. Nevertheless, there is no attempt to minimize the
number of bit flips for removing the forbidden configura-
tions by the 1-D (0, k)-RLL constraint from a given binary
codeword. Moreover, the main problem with the DBF method
introduced in [34] still is the number of deliberate bit errors
that may overwhelm the ECC decoder and affect the error-
floor performance (which limits its applications).
Our Contributions: One of the practical motivations to
design a DBF coding scheme for data-dependent channels
is to address the error propagation phenomena existing in
conventional 2-D constrained coding methods. Most of these
constrained coding schemes are non-linear, and their en-
coder/decoder has a memory such that over noisy channels
single channel bit errors may cause a decoder to lose track of
encoded bits and therefore propagate errors indefinitely with-
out recovering. On the other hand, the main problem with the
DBF method is the number of deliberate flips. This problem
becomes also much more difficult for the 2-D case, and it
is a challenge to design efficient algorithms for identifying
harmful configurations in channel input patterns, let alone
the problem of minimizing the number of bit flips which
may overwhelm the error-correction decoder. In this paper,
we reformulate the problem of minimizing the number of bit
flips in the DBF scheme for removing harmful configurations
from 2-D channel input patterns as a constrained combinatorial
optimization problem. Furthermore, we design a Generalized
Belief Propagation (GBP)-guided DBF algorithm for identi-
fying 2-D harmful configurations and removing them with
minimal number of flips. In order to use the GBP algorithm,
we present a probabilistic graphical model for the constrained
combinatorial minimization problem using the factor graph
formulation in [38], [39]. In this framework, patterns which
do not contain harmful configurations are assumed to be
uniformly distributed, and each pattern containing a harmful
configuration has zero probability. In this way, we reformulate
the problem as a 2-D maximum a posteriori (MAP) problem,
and demonstrate that the GBP algorithm can approximately
solve this 2-D MAP problem. In order to study and analyze
the performance of our proposed method, we introduce a
binary 2-D channel with memory which captures the effect
on an information bit from its surrounding patterns, i.e., the
neighboring bits. The channel is characterized by rules defined
by a set of configurations with a specific shape, which we call
the set of harmful configurations. At the channel output, the
probability of error for bits contained in any of the harmful
configurations are larger than for the other bits. We evaluate
the performance of the GBP-guided DBF method over the
introduced channel where the 2-D isolated-bits configurations
are considered as the channel harmful configurations. Further-
more, the performance of the DBF method for 2-D no isolated-
bits (n.i.b.) constraint on a memoryless binary symmetric
channel (BSC) is compared with the row-by-row and bit-
stuffing based 2-D n.i.b. encoders, presented in [10] and [40],
respectively.
(a) (b)
Fig. 1. Two examples of polyominoes: (a) a 2 × 2 square and (b) a cross.
Paper Organization: The rest of this paper is organized
as follows. Section II presents the notations and definitions
used throughout the paper. In Section III, the data-dependent
channel model is introduced. In Section IV, the problem of
minimizing the number of flipped bits in the DBF method
is formulated. In Section V, we reformulate the minimization
problem as a 2-D MAP problem, and explain the ideas of
using the GBP algorithm for solving this problem. Numerical
results are presented in Section VI. Section VII concludes the
paper.
II. NOTATIONS AND DEFINITIONS
We denote a discrete random variable with an upper case
letter (e.g., X) and its realization by the lower case letter (e.g.,
x). We denote the probability density function of X with p(x)
and the conditional probability density function of Y given
X by p (y|x). [n
1
: k : n
2
] represents the set of real numbers
{n
1
, n
1
+ k, n
1
+ 2k . . . , n
2
}, and [n] denotes [1 : 1 : n]. We
denote a random array of size m × n by X = [X
i,j
]
i∈[m],j∈[n]
.
An array of binary symbols with size m × n is denoted by x =
[x
i,j
]
i∈[m],j∈[n]
where x
i,j
∈ {0, 1} is the (i, j)
th
component
of array. A
m,n
=
(i, j) ∈ Z
2
: i ∈ [m] and j ∈ [n]
denotes
the index set of an array of size m × n and is the subset of the
2-D lattice Z
2
. The Hamming weight of an array x of binary
symbols is determined by w
H
(x) =
P
x
i,j
∈x
1
{x
i,j
= 1},
where
1
{.} equals one (respectively, zero) when its argument
is true (respectively, false). The XOR operation between two
binary arrays (x and y of size m × n) is done component-wise,
i.e., x⊕y = (z
i,j
)
i∈[m],j∈[n]
where z
i,j
= x
i,j
⊕ y
i,j
, and x
i,j
and y
i,j
are the (i, j)
th
component of x and y, respectively.
Furthermore, the Hamming distance between x and y is
determined by d
H
(x, y) = w
H
(x⊕y). A binary BCH code of
length N with N − K parity bits and minimum distance d
min
is denoted by BCH-[N, K, d
min
]. A binary Reed-Muller code
of length N = 2
m
with N − K = 2
m
−
P
r
i=0
m
i
parity bits
and minimum distance d
min
= 2
m−r
is denoted by RM-(r, m).
A polyomino of order k, called also a k-ominoe, is a plane
geometric figure formed by joining k neighboring square
shapes. Among polyominoes are 2 × 2 square-shaped poly-
ominoes
Q
(i, j) = {(i, j), (i, j + 1), (i + 1, j), (i + 1, j + 1)} , (1)
and cross-shaped polyominoes
Q
+
(i, j) = {(i, j − 1), (i − 1, j), (i, j), (i, j + 1), (i + 1, j)} ,
(2)
over the 2-D lattice Z
2
, which are shown in Fig. 1.
3
An m × n binary pattern is denoted by x = [x
i,j
]
i∈[m],j∈[n]
,
where x
i,j
indicates the value of bit in i-th row and j-th
column. Throughout the paper, white squares denote zero bits
and black squares represent 1. Consider a k-ominoe P and
the set of all 2
k
binary configurations of that shape X
P
. We
refer to them as to P-shaped configurations and denote them
by x
P
. As an example, Fig. 2 shows all binary configurations
of a 2 × 2 square-shaped polyomino.
Fig. 2. The set of all binary configurations of a 2 × 2 square-shaped
polyomino.
Consider x
i,j
over an m × n rectangular pattern x, then the
union of all P-shaped polyominoes that intersect with this bit
is denoted by P
i,j
. The configuration of P
i,j
is denoted by
x
P
i,j
. For the cases of 2 × 2 square-shaped and cross-shaped
polyominoes, we have
P
i,j
=
[
(i
0
,j
0
)∈Q
(i−1,j−1)
Q
(i
0
, j
0
), (3)
and
P
+
i,j
=
[
(i
0
,j
0
)∈Q
+
(i,j)
Q
+
(i
0
, j
0
), (4)
respectively. Fig. 3 shows P
i,j
for these polyominoes.
III. CHANNEL MODEL
In this section, we introduce a communication channel
transmitting binary rectangular patterns and producing as an
output a binary pattern. The channel is data-dependent and
characterized by rules defined by a set of binary configurations
of a P-shaped polyomino. We call this set of P-shaped con-
figurations the set of harmful configurations. At the channel
output, the error probability of bits contained in configurations
which belong to the set of harmful configurations is larger than
the other bits. Therefore, the channel has states and its error
statistics depends on input binary patterns. In the following,
we formally present error and state characterizations.
(a) (b)
Fig. 3. Figure demonstrates P
i,j
over a rectangle when the polyomino is:
(a) a 2 × 2 square and (b) a cross.
Fig. 4. 2-D isolated-bits patterns containing the bit x
i,j
.
The input and output alphabets X and Y are two sets
of binary rectangular patterns of size m × n. An m × n
binary pattern x = [x
i,j
]
i∈[m],j∈[n]
is chosen randomly and
uniformly from X as an input to the channel. The channel
output, y = [y
i,j
]
i∈[m],j∈[n]
∈ Y, is also a binary pattern of
size m × n. For x
i,j
, P
i,j
denotes the union of P-shaped
polyominoes that intersect with this bit, and x
P
i,j
is the
configuration of P
i,j
, as defined in Section II. We assume
that the set of all possible configurations for P
i,j
, denoted by
X
P
i,j
, can be partitioned into two disjoint subsets X
G
P
i,j
and
X
B
P
i,j
, i.e., X
P
i,j
= X
G
P
i,j
S
X
B
P
i,j
, where X
B
P
i,j
is the set of
configurations containing P-shaped configurations which are
harmful for the channel. For example, X
B
P
i,j
can be the set of
binary configurations of P
i,j
given in Fig. 3(b), which contains
the 2-D isolated-bit patterns. The 2-D isolated-bit patterns are
shown in Fig. 4.
For x
i,j
contained in a harmful P-shaped configuration, the
channel is in the bad state, and the probability of error is
α
b
. However, passing though the channel, a bit that does not
belong to a harmful configuration is in error with a probability
of α
g
, and the channel is in the good state. We assume that
α
b
α
g
, or, in other words, the probability of error for bits
contained in a harmful configuration is much larger than that
of the other bits. The received binary pattern is y = x ⊕ e
CH
,
where e
CH
= [e
CH
i,j
] is the channel error array. Therefore,
e
CH
i,j
has either Bernoulli(α
g
) or Bernoulli(α
b
) distribution,
depending on the pattern x
P
i,j
. In fact, the channel is a binary
symmetric channel (BSC) with crossover probability α
b
when
x
P
i,j
∈ X
P
B
i,j
and a BSC with crossover probability α
g
when
x
P
i,j
6∈ X
P
B
i,j
, respectively.
We define an indicator function for the channel
f
CH
: X
P
i,j
→ {0, 1} over every x
i,j
,
f
CH
x
P
i,j
=
1
n
x
P
i,j
∈ X
P
B
i,j
o
, (5)
to identify bits which are contained in harmful configurations,
where x
i,j
belongs to at least one harmful configuration if
f
CH
x
P
i,j
= 1. Using the above indicator function, we can
determine the channel state for transmission of x
i,j
as follows
s
i,j
=
(
b, f
CH
x
P
i,j
= 1,
g, f
CH
x
P
i,j
= 0,
(6)
where “b” and “g” stand for the bad and the good channel
states, respectively. Let the probability distribution function
of channel be p(y|x). According to the aforementioned error
characterization, the probability distribution function of chan-
nel can be factored into
p(y|x) =
Y
(i,j)
p
y
i,j
|x
P
i,j
, (7)
4
Fig. 5. A schematic representation for the channel model is given. Passing
through the channel, x
i,j
is in error with probability α
b
if the configuration of
P
i,j
, x
P
i,j
, belongs to the set of harmful patterns X
B
P
i,j
, otherwise it inverts
with a probability of α
g
. It should be noted that the top arm of the figure
can be removed when α
g
= 0, which reduces the channel into a constrained
2-D channel with the list of forbidden configurations X
B
P
i,j
. However, in our
channel removing the harmful patterns does not make the channel noiseless.
Removing all the harmful patterns in the set X
B
P
i,j
before transmission
through the channel, makes it a BSC with the cross-over probability α
g
.
since y
i,j
only depends on the configuration of P
i,j
in the
input pattern x. Fig. 5 gives a schematic illustration for the
channel.
Remark 1: In this paper, we use the concept of polyominoes
to just demonstrate the effect of harmful configurations on its
neighboring bits over a 2-D binary pattern. As two examples,
we consider 4-ominoes and 5-ominoes, as these reflect physi-
cal effects of 2-D ISI and ICI over the plane. For this purpose,
we defined the square and cross shaped polyominoes in (1) and
(2).
Remark 2: The channel is similar to the Gilbert-Elliot
channel [41], as it has two states, where each state acts
as a BSC with a different cross-over probability. However,
the state transitions in our channel model depend on input
patterns. For such channels, calculating the information rate,
let alone the capacity, is much more challenging than for
discrete memoryless channels. Except for very special cases,
there are no simple expressions for information rates available,
and so, one needs to rely on upper and lower bounds and/or
on stochastic techniques for estimating the information rate,
examples are [42]–[44].
Remark 3: The probability that the channel is in the bad
state (or, in the good state) depends on the input probability
distribution. If we assume that input bits are i.i.d., then
there is no Markovian assumption on the channel states. The
probability that the channel is in the bad state for sending x
i,j
is
p (s
i,j
= b) = p
f
CH
x
P
i,j
= 1
=
|X
B
P
i,j
|
|X
P
i,j
|
, (8)
as the patterns are chosen randomly and uniformly, and in the
good state is p (s
i,j
= g) = 1−p (s
i,j
= b). For different input
probability distributions, this probability can be computed
accordingly. Throughout the paper, we do not consider any
Markovian properties on input bits.
In the following, we present an example of an input binary
pattern to the channel, where the 2-D isolated-bits patterns are
the harmful patterns for the channel, to illustrate the effects
of harmful patterns on input binary patterns passing through
the channel.
Fig. 6. A 7 × 7 binary pattern x is transmitted through the channel with the
set of 2-D isolated-bits patterns as the set of harmful patterns. The bits x
2,6
,
x
3,5
, x
3,6
, x
3,7
, x
4,6
, x
6,7
, x
7,6
and x
7,7
belong to the 2-D isolated-bits
patterns. Passing through the channel, the probability of error for these bits
is α
b
, and for the rest of them is α
g
.
Example 1: Fig. 6 shows an example of a 7 × 7 input binary
pattern x transmitted over the introduced channel. We assume
that the set of harmful patterns for the channel is the set of
2-D isolated-bits patterns, which are given in Fig. 4. In order
to determine the channel state for all bits over the pattern, we
assume zero entries outside of x, i.e., x
i,j
= 0, while i < 1,
j < 1, i > 7, or j > 7. There are two isolated-bits patterns
in x, which are x
Q
+
(3,6)
and x
Q
+
(7,7)
. Passing through the
channel, the bits contained in these two harmful configurations
are in error with a probability of α
b
. These bits x
2,6
, x
3,5
, x
3,6
,
x
3,7
, x
4,6
, x
6,7
, x
7,6
and x
7,7
. For instance, for x
2,6
,
P
2,6
=
[
(i
0
,j
0
)∈Q
+
(2,6)
Q
+
(i
0
, j
0
). (9)
Since Q
+
(3, 6) ⊂ P
2,6
and x
Q
+
(3,6)
is a 2-D isolated-bits
pattern, we have the fact that x
P
2,6
contains a 2-D isolated-
bits pattern, and therefore, x
2,6
is in the bad state. Similarly,
we can check this for the rest of bits in x.
IV. PROBLEM FORMULATION
The user uniformly and randomly selects a binary
message m out of 2
K
messages denoted by M =
{m
1
, m
2
, . . . , m
2
K
}, where each message is of length K ∈ N.
The user message m is first encoded by an error correction
encoder with rate R =
K
N
. The error correction encoding
function φ
ECC
: M → S
N
ECC
assigns a binary codeword c(m)
of length N to the user data m such that
c(m) = φ
ECC
(m), (10)
where S
N
ECC
= {c(m
1
), c(m
2
), . . . , c(m
2
bNRc
)} is the code-
book (the set of binary codewords of length N) associated
with the ECC being used. A codeword c ∈ S
N
ECC
is represented
by N binary symbols, c = (c
1
, c
2
, . . . , c
N
), and N = m × n.
Each codeword is arranged into an array x of size m × n, such
that x = [x
i,j
]
i∈[m],j∈[n]
, and x
i,j
= c
(i−1)m+j
. The array
x can be considered as a binary rectangular pattern of size
m × n. We want to send the pattern x over the communication
channel in Section III, with the list of harmful configurations
X
B
P
i,j
. Assuming that α
b
α
g
, then bits contained in config-
urations of list X
B
P
i,j
are more prone to error than the other
bits. To overcome effects of harmful configurations, we use
a deliberate error insertion approach to remove the harmful
![Fig. 11. The average probability of error with and without incorporating for the cases (a) αg = 0 and αb ∈ [0.1 : 0.1 : 1], and (b) αg ∈ [0.001 : 0.001 : 0.01] and αb = 100× αg is presented. The BCH[1024, 728, 62], RM-(4, 10) and RM-(5, 10) codes are being used. The BER comparison results are obtained using the equations (33) and (34), and executing the GBP-guided DBF algorithm over at least 50,000 random instances of user messages.](/figures/fig-11-the-average-probability-of-error-with-and-without-35ow1r3m.webp)
![Fig. 12. Figure shows the BER comparison results of the DBF, bit-stuffing and row-by-row coding methods on the BSC with the cross-over probability (α). The effect of error propagation can be observed in the BER curve of bit-stuffing which shows that this method is vulnerable to channel errors. The coding rate of DBF with BCH-[1024, 923, 22] code is close to the bit-stuffing method, and the rate of DBF with BCH-[1024, 768, 54] is close to the rate of row-by-row coding method.](/figures/fig-12-figure-shows-the-ber-comparison-results-of-the-dbf-sskl25ll.webp)
