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Conference Paper

Efficient Design of Capacity-Approaching Two-Dimensional Weight-Constrained Codes

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In this work, given n, p>0 , efficient encoding/decoding algorithms are presented for mapping arbitrary data to and from n×n binary arrays in which the weight of every row and every column is at most pn. Such constraint, referred as p-bounded-weight-constraint, is crucial for reducing the parasitic currents in the crossbar resistive memory arrays, and has also been proposed for certain applications of the holographic data storage. While low-complexity designs have been proposed in the literature for only the case p=1/2 , this work provides efficient coding methods that work for arbitrary values of p . The coding rate of our proposed encoder approaches the channel capacity for all p .
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Efﬁcient Design of Capacity-Approaching
Two-Dimensional Weight-Constrained Codes
Tuan Thanh Nguyen, Kui Cai, Kees A. Schouhamer Immink, and Yeow Meng Chee
Singapore University of Technology and Design, Singapore 487372
Turing Machines Inc, Willemskade 15d, 3016 DK Rotterdam, The Netherlands
Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore
Emails: {tuanthanh nguyen,cai kui}@sutd.edu.sg, immink@turing-machines.com, pvocym@nus.edu.sg
Abstract—In this work, given n, p > 0, efﬁcient encoding/decoding
algorithms are presented for mapping arbitrary data to and from
n×nbinary arrays in which the weight of every row and every
column is at most pn. Such constraint, referred as p-bounded-
weight-constraint, is crucial for reducing the parasitic currents in
the crossbar resistive memory arrays, and has also been proposed
for certain applications of the holographic data storage. While low-
complexity designs have been proposed in the literature for only
the case p= 1/2, this work provides efﬁcient coding methods that
work for arbitrary values of p. The coding rate of our proposed
encoder approaches the channel capacity for all p.
I. INTRODUCTION
A solution for the ever-increasing demand for data storage
capacity is to scale the storage density while maintaining its reli-
ability. Particularly, in the resistive memory crossbar arrays or the
holographic recording systems, the information data is regarded
as two-dimensional (2D) instead of one-dimensional (1D). Cor-
respondingly, new types of 2D constraints are introduced into
the input information data to improve the system reliability. In
this work, we focus on constraint, namely 2D bounded-weight-
constraint. We next brieﬂy describe the applications of such a
constraint in practical data storage systems.
In resistive memory, the memory cell is a passive two-terminal
device that can be both read and written over a simple crossbar
structure [1]–[6], which facilitates a huge density advantage.
However, a fundamental and challenging problem of the crossbar
memory arrays is the sneak path problem. When a cell in a
crossbar array is read, a voltage is applied upon it, and current
measurement determines whether it is in a low-resistance state
(LRS, corresponding to a ‘1’) or a high-resistance state (HRS,
corresponding to a ‘0’). Sneak paths are an effect by which
in parallel to the desired measurement path, alternative current
paths through other array cells distort the measurement, which
may result in reading an erroneous state. The sneak path problem
was addressed by numerous works with different approaches and
at various system layers [7]–[11]. An effective method to reduce
the sneak path effect is by enforcing fewer memory cells with
the LRSs. This can be achieved by applying constrained coding
techniques to convert the user data into 2D-constrained array
that limits number of 1’s in every row and every column. For
example, Ordentlich and Roth [7] required the weight in every
row and every column to be at most half, and presented efﬁcient
encoders with redundancy at most 2nfor n×narray. In [12],
the authors studied the bounds of codes that required the weight
in every row and every column is precisely pn and provided
a coding scheme based on the enumeration coding technique.
In this work, we provide efﬁcient designs for n×nbinary
arrays that satisfy the p-bouned-weight-constraint, in which the
weight in every row and every column is at most pn for
arbitrary p(0,1). Besides resistive memory, this constraint
has been proposed for applications in holographic memory as
well [12], [13]. Particularly, Talyansky et al. [13] studied the t-
conservative arrays constraint, where every row (not necessary
column) has at least ttransitions 01or 10for some
t6n/(2 log n)O(log n). It is easy to verify that such
constraint is equivalent to the p-bounded-weight constraint for
p= 1 1/(2 log n)>1/2in a weaker condition, where the
weight-constraint is enforced in the rows only. We show that
our encoders can be used for such a constraint with a strictly
larger range of t(refer to Section III, Remark 1).
While 1D-constraints have been extensively investigated, the
study for 2D-constraints is much less profound. The main reason
is due to the provable difﬁculty of 2D-constraints compared to
1D-constraints. For example, consider certain weight-constrained
codes such as the balanced codes or constant-weight codes, there
are several efﬁcient prior-art coding methods for designing 1D-
codes with optimal or almost optimal redundancy [14]–[17].
Here, almost optimal refers to the cases that the encoder’s
redundancy is at most a constant bit away from the optimal re-
dundancy. Consider such constraints in 2D-codes, although low-
complexity encoding algorithms have been introduced, there is
still a big gap between the corresponding encoder’s redundancy
and the optimal value.
For 2D p-bounded-weight-constraint, one may use the method
proposed by Ordentlich and Roth [12] for arbitrary p(0,1) or
the recent method proposed by Nguyen et al. [21] for p61/2.
Both methods are based on the enumeration coding technique,
that resulted in high complexity encoding/decoding. The main
contribution of this work is to further reduce the redundancy of
previous known encoders for p-bounded-weight-constraint while
having a lower complexity of encoding/decoding algorithm. Our
coding scheme works for arbitrary values of p(0,1) and the
rates of our proposed encoders approach the channel capacity.
The coding method can be easily modiﬁed to construct m×n
arrays.
We ﬁrst go through certain notations and review prior-art
coding techniques.
II. PRELIMINARIES
A. Notations
Given two binary sequences x=x1. . . xmand y=y1. . . yn,
the concatenation of the two sequences is deﬁned by xy ,
x1. . . xmy1. . . yn.For a binary sequence x, we use wt(x)to
denote the weight of x, i.e the number of ones in x. We use x
to denote the complement of x. For example, if x= 00111 then
wt(x) = 3 and x= 11000.
Let Andenote the set of all n×nbinary arrays. The ith row of
an array AAnis denoted by Aiand the ith column is denoted
by Ai. We use Ai;hjito denote the sequence obtained by taking
the ﬁrst jentries of the row Aiand use Ai;hjito denote the
sequence obtained by taking the ﬁrst jentries of the column Ai.
Given n, p > 0, we set
B(n, p) = nAAn: wt(Ai)6pn and wt(Ai)6pn for 16i6no.
The channel capacity is deﬁned by
cp,lim
n→∞
log |B(n, p)|
n2.
In this work, we are interested in the problem of designing
efﬁcient coding methods that encode (decode) binary data to
(from) B(n, p).
Deﬁnition 1. The map EN C :{0,1}k→ {0,1}n×nis a p-
bounded-weight-constrained encoder if ENC(x)B(n, p)for all
x∈ {0,1}kand there exists a decoder map DEC :{0,1}n×n
{0,1}ksuch that DEC ENC (x) = x. The redundancy of the
encoder is measured by the value n2k(bits).
Our design objectives include low redundancy (equivalently,
high code rate) and low complexity of the encoding/decoding
algorithms. In this work, the rates of our constructive codes
approach the channel capacity for all values of p. Particularly,
for p>1/2, we have cp= 1.
B. Previous Works
We brieﬂy describe the works on the p-bounded-weight
constraint by Ordentlich and Roth in [7], [12] that provided
encoding/decoding algorithms for B(n, p).
For p= 1/2, i.e. the weight of every row and every
column is at most n/2. In [7], the authors presented two low
complexity coding methods. The ﬁrst method is based on
ﬂipping rows and columns of an arbitrary binary array until
the weight-constraint is satisﬁed in all rows and columns
while the second method is based on efﬁcient construction
of antipodal matching. Both codes have roughly 2nredun-
dant bits. A lower bound on the optimal redundancy was
shown to be λn +o(n)for a constant λ1.42515 in
[20]. Note that these two methods can be used to construct
B(n, p)for arbitrary p > 1/2since B(n, 1/2) B(n, p)
for all p > 1/2.
For p < 1/2, one may follow the coding method in
[12], based on enumeration coding, that ensure the weight
in every row and every column to be precisely pn. The
redundancy of the proposed encoder was at most 2(n, p),
where µ(n, p)is the least redundancy required to encode
one-dimensional binary codewords of length nsuch that
the weight is pn. If we set
Q(n, p) = nx∈ {0,1}n: wt(x) = pno,
then we have µ(n, p) = nH(p)log |Q(n, p)|where
H(p) = plog p(1 p) log(1 p). It is easy to verify
that µ(n, p) = Θ(n)for all p < 1/2. In addition, Ordentlich
and Roth [12] also provided a lower bound on the optimal
redundancy, which is at least 2(n, p)+O(n+ log n)bits.
In this work, we propose two efﬁcient coding methods for
B(n, p). The ﬁrst method combines the sequence replacement
technique and antipodal matching to encode B(n, p)for arbitrary
p > 1/2with n+ 3 redundant bits. On the other hand, the
second method, based on Knuth’s balancing technique, uses at
most (n, p) + O(nlog n)redundant bits to construct B(n, p)
for arbitrary p < 1/2. We review below the antipodal matching
(deﬁned in [7]) and the linear-time encoder for weakly-balanced
binary codes [18], as they will be used in our proposed 2D
constrained coding methods.
Deﬁnition 2 (Ordentlich and Roth [7]).An antipodal matching φ
is a mapping from {0,1}nto itself with the following properties
holding for every x∈ {0,1}n:
1) wt(φ(x)) = nw(x).
2) If wt(x)> n/2then φ(x)has all its 1’s in positions where
xhas 1’s. In other words, suppose x=x1x2. . . xnand
y=φ(x) = y1y2. . . yn, then yi= 1 implies xi= 1 for
16i6n.
3) φ(φ(x)) = x.
Ordentlich and Roth [7] presented an efﬁcient construction
of antipodal matchings φfor all n. In fact, such an antipodal
matching can be decomposed into a collection of bijective
mappings φ=n
i=0φiwhere
φi:nx∈ {0,1}n: wt(x) = ionx∈ {0,1}n: wt(x) = nio.
The authors showed that φican be constructed in linear-time
for all n, i.
In [18], we studied the weakly-balanced constraint that en-
forces the weight-constraint over every window of size =
Ω(log n), and showed that for arbitrary p1, p2, where 06
p1<1/2< p261, for sufﬁcient n, there exists a linear-
time algorithm to encode binary data to codewords of length
nwith only 1 redundant bit. Our coding method is based on
the sequence replacement technique and the complexity of the
algorithm was shown to be linear in codeword’s length [18].
Theorem 1 (Nguyen et al. [18]).Given p1, p2where 06
p1<1/2< p261, let c= min{1/2p1, p21/2}.
For (1/c2) logen66n, there exists linear-time algorithm
ENC :{0,1}n1→ {0,1}nsuch that for all x∈ {0,1}n1if
y=ENC(x)then wt(y)[p1n, p2n]) and for every window w
of size of y,wt(w)[p1, p2].
III. CONSTRUCTION OF B(n, p)FOR p > 1/2
Recall that the encoders proposed in [7] can be used for
constructing B(n, p)when p > 1/2, and the redundancy is
roughly 2n(bits). In this section, we provide a linear-time
encoder for B(n, p)where p > 1/2with at most (n+ 3)
redundant bits.
Set m=n2(n+3),=n,p1= 0,p2=pand c=p1/2.
According to Theorem 1, for sufﬁcient nthat (1/c2) loge(n2
n3) 6n6n2n3, there exists linear-time encoder,
ENCseq :{0,1}m→ {0,1}m+1 such that for all x∈ {0,1}m
and y=ENCseq (x)we have wt(w)[0, pn]for every window
wof size nof y. In addition, we follow [12] to construct the
antipodal matchings φfor sequences of length n1.
We now describe the detailed construction of the p-bounded-
weight-constrained encoder, ENCB(n,p). We have p > 1/2in all
our descriptions in this part.
p-bounded-weight-constrained encoder, ENCB(n,p).
INP UT:x∈ {0,1}m
OUTPUT:A,ENCB(n,p)(x)B(n, p)with p > 1/2
(I) Set y=EN Cseq(x)∈ {0,1}m+1 . Suppose that y=
y1y2. . . yn2n2.
(II) Fill n2n1bits of yto Arow by row as follows.
Set Ai,yn(i1)+1 . . . yni for 16i6n2.
Set An1,yn(n2)+1 . . . yn2n212, where 1,2
are determined later.
Suppose that An=z1z2. . . znwhere ziis determined
later.
If wtAn1;hn2i> p(n2), ﬂip all bits in
An1;hn2iand set 1= 1, otherwise set 1= 0.
(III) For 16i6(n1), we check the ith column:
if wtAi;hn1i> pn, set zi= 1 and replace
Ai;hn1iwith φAi;hn1i
Otherwise, set zi= 0.
(IV) Check the nth row:
If wtAn;hn1i> pn, set 2= 1 and replace
An;hn1iwith φAn;hn1i
Otherwise, set 2= 0.
(V) Check the nth column:
If wtAn;hn1i> pn, set zn= 1 and replace
An;hn1iwith φAn,hn1i.
Otherwise, set zn= 0.
(VI) Output A.
Theorem 2. The Encoder ENCB(n,p)is correct. In other words,
ENCB(n,p)(x)B(n, p)with p > 1/2for all x∈ {0,1}m. The
redundancy is n+ 3 (bits).
Proof. Let A=ENCB(n,p)(x). We ﬁrst show that the weight of
every column of Ais at most pn. From Step (III) and Step (V),
the encoder guarantees that the weights of ncolumns are at most
pn. Although there is a replacement in the nth row, it does not
affect the weight of any column. Indeed, if wt(x)> n/2, then
φ(x)has all its 1’s in positions where xhas 1’s and wt(φ(x)) 6
n/2. Therefore, whenever the encoder performs replacement step
in any row (or respectively any column), it does not increase the
weight of any column (or respectively any row). Therefore, we
conclude that wtAi6pn for all 16i6n.
We now show that the wtAi6pn for all 16i6n.
From Step (IV), we observe that the nth row satisﬁes the
weight-constraint. As mentioned above, during Step (III) and
Step (V), whenever the encoder performs replacement step in
any column, it does not increase the weight of any row, i.e. the
weight-constraint is preserved over the ﬁrst (n2) rows that
is guaranteed by the Encoder ENCseq from Step (I). It remains
to show that the (n1)th row satisﬁes the weight constraint
with the determined values of 1,2. Indeed, from Step (II), if
1= 0, we have:
wtAn16p(n2) + 1 = pn + (1 2p)6pn.
Otherwise,
wtAn1<(1 p)(n2) + 2 < pn for all n > 2/(2p1).
In conclusion, we have ENCB(n,p)(x)B(n, p)for all x
{0,1}m. Since m=n2(n+3), the redundancy of our encoder
is n+ 3 (bits).
For completeness, we describe the corresponding decoder
DECB(n,p)as follows.
p-bounded-weight-constrained decoder, DECB(n,p).
INP UT:AB(n, p)
OUT PU T:x,DEC B(n,p)(A)∈ {0,1}m, where m=n2n3
(I) Decode the nth column, An. If the last bit is 1, ﬂip it
to 0 and replace An;hn1iwith φAn;hn1i. Otherwise,
proceed to the next step.
(II) Decode the nth row, An. Check the last bit in An1, if it
is 1, ﬂip it to 0 and replace An;hn1iwith φAn;hn1i.
Otherwise, proceed to the next step.
(III) Decode the (n1)th row, An1. Check the second last
bit in An1, if it is 1, ﬂip all the bits in An1. Otherwise,
proceed to the next step.
Suppose the nth row is now An=z1z2. . . zn.
(IV) For 16i6(n1), we decode the ith column, i.e. Ai,
as follows. If zi= 1, replace Ai;hn1iwith φAi,hn1i.
(V) Set y,A1A2. . . An2An1;hn2i∈ {0,1}n2n2.
(VI) Output x,DE Cseq(y)∈ {0,1}n2n3.
Complexity analysis. For n×narrays, it is easy to verify
that Encoder ENCB(n,p)and Decoder DE CB(n,p)have linear-time
complexity. Particularly, there are at most n+ 2 replacements,
each replacement is done over sequence of length n, and the
complexity of encoder/decoder EN Cseq,DECseq is linear over
codeword length m=n2n3 = Θ(n2). We conclude that
the running time of Encoder ENCB(n,p)and Decoder D EC B(n,p)
is Θ(n2)which is linear in the message length m=n2n3.
Remark 1. In [13], Talyansky et al. studied the t-conservative
arrays constraint, where every row has at least ttransitions 0
1or 10for some t6n/(2 log n)O(log n)< n/2. Such a
constraint is equivalent to the p-bounded-weight constraint in a
weaker condition, where the weight-constraint is enforced in the
rows only. Indeed, for a sequence x=x1x2. . . xn∈ {0,1}n,
consider the differential of x, denoted by Diﬀ(x), which is a
sequence y=y1y2. . . yn∈ {0,1}n, where y1=x1and yi=
xixi1(mod 2) for 26i6n. We then observe that x
has at least ttransitions if and only if the weight of Diﬀ(x)is
at least t. In addition, the constraint problem where the weight
in every row is at least twhere t < n/2is equivalent to the
constraint problem where the weight in every row is at most t
where t > n/2. Therefore, one may modify the construction of
our proposed encoder ENCB(n,p)(i.e for p= 1 1/(2 log n)) >
1/2) to construct binary arrays such that there are at least t
transitions in every row and every column with at most n+ 3
redundant bits. When the weight-constraint is only required on
rows, only Step (I) in EN CB(n,p)is sufﬁcient and m=n2
1. Although there is no improvement in the redundancy (the
encoder in [13] also use only one redundant bit), our encoder
can be applied for a larger range of t, where t6n/c for any
c > 2.
IV. CONSTRUCTION OF B(n, p)FO R p61/2
The sequence replacement method to construct EN Cseq ,
DECseq is efﬁcient in terms of redundancy (only one bit)
and running time complexity (linear-time). However, it is not
applicable to the cases when p61/2. For such cases, we
adapt the enumeration coding technique and a modiﬁcation of
the Knuth’s balancing technique [15]. Compare to literature
works in [12], [21] that also used modiﬁcations of enumeration
coding technique, the major difference of our coding method is
that the rows and columns are encoded independently, and the
two-dimensional code construction can be divided to two one-
dimensional code constructions. In other words, the complexity
of our encoder mainly depends on the efﬁciency of enumeration
coding for one dimensional codes.
In general, a ranking function for a ﬁnite set Sof cardinality N
is a bijection rank : S[N]where [N],{0,1,2, . . . , N 1}.
Associated with the function rank is a unique unranking function
unrank : [N]S, such that rank(s) = jif and only if
unrank(j) = sfor all sSand j[N]. Given n, p > 0,
let S(n, p),nx∈ {0,1}n: wt(x)6pno, i.e. S(n, p)is
the set of all one dimensional sequences that satisfy the p-
bounded-weight constraint. One may use enumeration coding
[22], [23] to construct rankp:S(n, p)[|S(n, p)|]and
unrankp: [|S(n, p)|]S(n, p). The redundancy of this
encoding algorithm is then λ(n, p) = nlog |S(n, p)|(bits).
A. Construction of B(n, p)for p < 1/2
We now present efﬁcient encoding method for B(n, p)for
given p < 1/2. Recall that one may follow the coding method
in [12], based on enumeration coding, that ensure the weight in
every row and every column to be precisely pn. The redundancy
of the proposed encoder was at most 2(n, p), where µ(n, p)is
the least redundancy required to encode one-dimensional binary
codewords of length nsuch that the weight is pn (refer to
Section II-B). It is easy to verify that λ(n, p)6µ(n, p). The
redundancy of our encoder is at most (n, p) + O(nlog n)6
(n, p)+ O(nlog n)bits. Our method is based on the enumera-
tion coding [22], [23] used to enforce the weight-constraint in all
rows (here rows are encoded/decoded independently) and swap,
used to enforce the weight-constraint in all columns (without
changing the weight in any row). The swap method is a simple
modiﬁcation of Knuth’s balancing technique [15], which was
also introduced in [13] for balanced codes.
We ﬁrst deﬁne p-bouned-weight-constraint in a general way.
Let xbe a q-dimensional binary vector (q>1) contains a
total of nbits. We say xis p-bounded if the weight of x
(also deﬁned by the number of 1’s in x) is at most pn. For
simplicity, we assume that pn, log n, (1/p)are integers. Set
m=n(1/p)(1 + log n), N =m(nλ(n, p)). We now
describe brieﬂy the main idea of the algorithm. The encoding
method includes three phases.
In Phase I, the encoder encodes the information of length N
into array Aof size m×nwhere every row is p-bounded,
using the enumeration coding technique. Particularly, the
information is encoded into S(n, p)with the redundancy at
most λ(n, p)bits for each row. Therefore, the redundancy
used in Phase I is at most (n, p)bits.
In Phase II, the encoder follows the swapping procedure
to ensure that every column of Ais p-bounded. Note that,
from Phase I, array Ais p-bounded. Suppose that at some
encoding step i, we have an array Sof size m×n0, which
is already p-bounded for some n06n(initially, n0n).
We then divide Sinto two subarrays of size m×(n0/2),
denoted by LSand RS, and proceed to make sure LSand
RSare both p-bounded. This can be done via a simple
method of swapping the bits in LSand RSas follows.
Swapping Procedure. If both LSand RSare p-bounded
then no action is needed, i.e. the number of swap is 0.
On the other hand, if one of them is not p-bounded, and
w.l.o.g, assume that wt(LS)> pn0while wt(RS)< pn0
where n0=mn0/2. We then swap the bits in LSand RS
respectively in the order from column to column. Observe
that each swap increases or decreases the weight of LS, RS
by one, and when all the bits in LSand RSare swapped,
we have wt(LS)< pn0while wt(RS)> pn0. Therefore,
there must be an index t,16t6n0such that after
swapping tbits, we have wt(LS)< pn0and wt(RS)< pn0.
We refer such an index as a swapping index of LSand
RS. To represent the swapping index t, we need at most
log n0redundant bits. After both subarrays LSand RS
are p-bounded, we continue to divide each of them into
two subarrays and repeat the swapping procedure to ensure
the newly created subarrays are p-bounded. This process
ends when all subarrays of size m×1are p-bounded. We
illustrate the idea of the swapping procedure in Figure 1.
Let re(n)be the sequence obtained by concatenating all
binary representations of the swapping indices. The size of
re(n), is at most
X
k=2j
26k6n
(n/k) log(mk/2) = (n1)(1 + log m)log n
< n(1 + log n)(bits).
In Phase III, the encoder encodes re(n)into an array Bof
size (nm)×nsuch that its every row and every column is
p-bounded. At the end of Phase III, the encoder outputs the
concatenation of Aand B, which is an array of size n×n.
Recall that nm= (1/p)(1 +log n). While the encoder in
[13] proceeds to repeat the swapping procedure to encode
re(n), which makes the problem difﬁcult to construct en-
coder for arrays of given size, we show that re(n)can be
encoded/decoded efﬁciently without repeating the swapping
procedure. Suppose that re(n) = x1x2. . . xn(1+log n),re(n)
is encoded to Bas follows. Here 0tdenotes the repetition
of 0for ttimes.
r1
B= (x10
1
p
1)(xnm+10
1
p
1). . . (x(nm)(n1)+10
1
p
1)
r2
B= (0x20
1
p
2)(0xnm+20
1
p
2). . . (0x(nm)(n1)+20
1
p
1)
. . .
ri
B= (0j1xi0
1
p
j). . . (0j1x(nm)(n1)+i0
1
p
j),
where ji(mod 1/p)and 16i6nm. It is easy to
verify that every row and every column of Bis p-bounded
and re(n)can be decoded uniquely from B.
In conclusion, the total redundancy for encoding an n×n
array is at most
(n, p) + (1/p)n(1 + log n)< nλ(n, p) + O(nlog n)bits
< nµ(n, p) + O(nlog n)bits.
Since µ(n, p) = Θ(n), we observe that the code rate of our
proposed encoder approaches the channel capacity. In general,
when pn,log nand 1/p are not integers, we can easily modify
the construction of our encoder and show that the redundancy
is at most (n, p) + O(nlog n)6(n, p) + O(nlog n)bits.
Due to space constraint, we defer the detailed construction for
the general case in the full paper.
Fig. 1: Swapping procedure example for n= 8, p = 1/4. The
current subarray Sis of size 8×4. The subarrays, highlighted
in red, are not p-bounded while those, highlighted in blue, are
p-bounded.
B. Improvement from Literature Works for p= 1/2
While the coding method in Section IV-A can also apply for
p= 1/2, the redundancy of the corresponding encoder will be
higher than literature works [7]. When p= 1/2, we can combine
the enumeration coding technique with the antipodal matching
to further reduce the redundancy of the proposed encoders in [7]
for even nas follows.
For an even n, set N=|S(n, 1/2)|=Pn/2
i=0 n
i. Suppose
that rank : S(n, 1/2) [N]and unrank : [N]S(n, 1/2) are
constructed. Given tbinary sequences x1,x2,...,xt, of length
n, we use x1||x2|| . . . ||xtto denote the t×narray where the ith
row is xi.
To encode B(n, 1/2), we ﬁrst ensure that the constraint is en-
forced over every row. Particularly, the information is encoded in
the ﬁrst (n2) rows using enumeration coding, the information
in the (n1)th is done via simple ﬂipping (we also reserve the
last two bits in this row as redundant bits), and the entire last
row is reserved for redundant bits.
For x∈ {0,1}(n2) log N, suppose d[Nn2]that takes xas
the binary representation of length (n2) log N, we represent d
in N-ary representation as d=Pn2
i=1 di×Ni1. Suppose that
ri
A= unrank(di), we then set EN C(x) = r1
A||r2
A|| . . . ||rn2
A.
For 16i6n2, if rank(ri
A) = diwhere di[N], then
DEC(r1
A||r2
A|| . . . ||rn2
A) = rank(d1)rank(d2). . . rank(dn2).
Recall that the right hand side is the concatenation of (n2)
sequences of length log N. We are now ready to present the
construction of ENCB(n,1/2) ,DECB(n,1/2) . The construction is
similar to the construction of B(n, p)for p > 1/2.
Encoder, EN CB(n,1/2).
Preparation phase. Given n > 0, n is even, set N=
|S(n, 1/2)|, and m= (n2) log N+ (n2).
INP UT:x∈ {0,1}m
OUT PU T:A,ENC B(n,1/2)(x)B(n, 1/2)
(I) Set x1be the preﬁx of length (n2) log Nof xand x2
be the sufﬁx of length n2of x.
(II) Set B=EN C(x1)which is a (n2) ×narray.
(III) For 16i6n2set Aibe the ith row of B.
(IV) Set An1=x212where 1,2∈ {0,1}that are
determined later.
If wt(x2)>1/2(n2), we set 1= 1 and ﬂip every
bit in x2, i.e. replace x2with the complement x2.
Otherwise, set 1= 0.
Suppose that An=z1z2. . . zn.
(V) For 16i6(n1), we check the ith column:
if wtAi;hn1i>1/2(n1), set zi= 1 and replace
Ai;hn1iwith φAi;hn1i
Otherwise, set zi= 0.
(VI) Check the nth row:
If wtAn;hn1i>1/2(n1), set 2= 1 and replace
An;hn1iwith φAn;hn1i
Otherwise, set 2= 0.
(VII) Check the nth column:
If wtAn;hn1i>1/2n, set zn= 1 and replace
An;hn1iwith φAn;hn1i.
Otherwise, set zn= 0.
(VIII) Output A.
V. CONCLUSION
We have presented efﬁcient encoding/decoding algorithms for
mapping arbitrary binary data to binary arrays of size n×nthat
satisfy the p-bounded-weight constraint, i.e. the weight in every
row and every column is at most pn. Our proposed encoders have
low complexity, and the rate approaches the channel capacity for
all values of p(0,1).
ACKNOWLEDGMENT
The work of Kui Cai and Tuan Thanh Nguyen is supported
by Singapore Ministry of Education Academic Research Fund
MOE2019-T2-2-123. The research of Yeow Meng Chee is
supported by the Singapore Ministry of Education under grant
MOE2017- T3-1-007.
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