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Flash Code with Dual Modes of Encoding
Michael Joseph Tan1, Proceso Fernandez2, Nino A. Salazar2, Jayzon Ty2, and
Yu ic hi K a ji 1
1Graduate School of Information Science
Nara Institute of Science and Technology,
8916–5 Takayama, Ikoma, Nara, 630–0192 Japan
{joseph-t,kaji}@is.naist.jp
http://http://www.naist.jp/en/
2Department of Information Systems and Computer Science
Ateneo de Manila University,
Loyola Heights, Quezon City, 1108 Philippines
{pfernandez}@ateneo.edu,{oh_ninja_nas,jayzonkid_21}@yahoo.com
http://http://www.admu.edu.ph
Abstract. This paper proposes a novel coding scheme which can extend
the lifespan of flash memory. Flash memory has a number of advantages
against conventional storage devices, but it must be noted that flash
cells which constitute a flash memory accommodate, not a small, but
limited number of operations only. A flash code provides a clever way to
represent data values in flash memory so that the number of operations
over flash cells become as small as possible, which contributes to extend
the lifespan of flash memory. Several flash codes have been studied so
far, and this paper proposes a novel coding scheme which makes use of
two di↵erent modes of encoding. Computer simulation shows that the
proposed coding scheme shows much better average-case performance
than existing codes. Besides the computer simulation, the paper also
gives detailed analysis of the performance which justifies the advantage
of the proposed code from a more theoretical viewpoint.
Keywords: flash memory, flash code, binary-indexed, dual-mode en-
coding
1 Introduction
Flash memory is a non-volatile memory that consists of an array of floating-gate
cells which are grouped in uniform sized blocks. Each cell can store at most
q1 levels of electric charge which is represented as an integer value. A peculiar
characteristic of a flash memory is the asymmetric relationship between the two
operations over the cell values. It is simple to raise the level of one cell, but due
to physical constraints, reversing the procedure is not allowed. The only way to
lower a cell level is through a block erasure which completely removes the charges,
bringing the cell level to zero, in an entire erase block of flash memory cells. The
continuous use of a block erasure is not favorable because it is slow and inflicts
2 Flash Code with Dual Modes of Encoding
physical damages to the cells. Practical flash memory often employs a wear-
leveling mechanism which contributes to extend the lifespan of flash memory by
equalizing the damages of cells evenly. Despite these e↵orts, the limited lifespan
is still a significant issue of flash memory.
One option to extend the lifespan of a flash memory is to improve the coding
scheme that updates and interprets the data stored in a flash memory. Such
coding scheme is called flash code and it is the target of this study. Flash codes
can be considered as a generalization of coding scheme for write-once memories
and this coding scheme have been studied since 1980s [1][7]. The framework for
flash code itself was only introduced in 2007 [3]. In this framework, we consider
to store a k-bit data in an erase block which consists of nflash cells, and the
performance of a flash code is evaluated in terms of a write deficiency which can
be regarded as a quantitative measure of the overhead of the code. For small
nand k, Jiang et al. proposed good flash codes which give more number of
operations (and hence less write deficiency) than naive simple coding scheme[3].
In [4], investigations are made to enlarge the parameters nand k. Mahdavifar
et al. improved the idea of [4], and proposed the index-less indexed flash code
(ILIFC)[6]. In ILIFC, cells in an erase block are grouped into smaller sub-blocks
which we call slices. Each bit of the k-bit data is assigned with one slice, and the
slice is used to remember the current value of the corresponding bit. ILIFC is a
simple and powerful coding scheme, but the problem with ILIFC is that the size
of a slice must be kor more. This restriction degrades the flexibility of the code
construction, and furthermore, a↵ects the performance of the code in a direct
manner.
There are two area of discussions with regard to the performance of flash
code. In papers such as [3, 4, 6], they discuss on lowering the worst-case write
deficiency of flash codes. Other papers such as [2, 5] discuss the importance
of constructing flash code that has a low write deficiency in the average-case
scenario. Both scenarios are equally important. The worst-case scenario depicts
the durability of the flash code. On the other hand, the average-case scenario
shows the practicality of a flash code which should also be considered since flash
memory are intended to be mass produced and be used by people in their daily
lives.
Based on the above described observations, the authors have proposed sev-
eral flash codes which can be regard as improvements of ILIFC [8,9]. Those
codes surely show smaller write deficiency than ILIFC, but the use of slices in
constructing flash codes seems to bring a certain overhead. To mitigate the issue
of using slices, we propose in this paper a new approach in constructing flash
code by integrating two modes of encoding. Each mode of encoding is a flash
code on its own, but by integrating them we are able to capitalize on their advan-
tages and mitigate their disadvantages. The proposed code shows a much smaller
write deficiency than existing flash codes in terms of average-case discussion, and
allows accommodating a large data in one fixed size block.
Flash Code with Dual Modes of Encoding 3
2 Preliminaries
Ablock in a flash memory consists of some fixed number nof cells, and this can
be represented by a vector C=(c0,c
1,...,c
n1)2An
qwith Aq={0,...,q1}.
Each element ciof Cindicates the amount of charge of the corresponding cell.
A cell with a charge 0 is said to be empty while a cell with a charge q1 is said
to be full. A cell that is neither empty nor full is said to be active. This concept
is extended to groups of cells which are analogously classified as empty (if it is
(0,...,0)), full (if it is (q1,...,q1)) or active (neither empty nor full). For
two states C=(c0,...,c
n1) and C0=(c0
0,...,c
0
n1), we write CC0if cic0
i
for 0 i<nand CC0if CC0and C6=C0. A state can transit from Cto
C0only if CC0.
The information stored in a single block is a k-bit data D=(d0,d
1,...,d
k1),
where di2{0,1}and kn. The value of the data is updated through a write
operation which inverts the binary value of a single bit in the data D. A flash code
Fbridges together the physical state of a block with the logical interpretation
of the data that the block contains.
Formally, a flash code consists of a pair of functions F=(E,D). The encoding
function E:{0,1,...,k1}⇥An
q!An
q[{E}is invoked at each write operation,
and E(j, C) provides the rules on how to write a new state (possibly a block
erasure E) to the block given the index jof the data bit djto be changed and
the current state C2An
qof the block. The decoding function D:An
q!{0,1}k
interprets the contents of the block into the corresponding k-bit data value.
Furthermore, if E(j, C )=C0where C06=Eand jis the index of the bit given
by a write operation, then value of D(C) and D(C0) are the same except for j-th
bit position.
A flash code increases at least one cell level for each write operation, and
therefore, the maximum number of write operations allowed for a block is n(q
1).This value is used to define a metric for flash codes, called the write deficiency:
(F)=n(q1) t(1)
where tis the number of write operations that the flash code Fis able to accom-
modate before a block erasure. Flash codes are usually compared against one
another in both the average-case and worst-case performances using the write
deficiency metric, and a smaller value of (F) is more favorable. To normal-
ize this metric, we further divide the write deficiency by n(q1), to produce
the write deficiency percentage which we use for the average case performance
analysis in later sections.
Among several constructions of flash codes, the authors are especially in-
terested in those suitable for large parameters of nand k. In this aspect, the
index-less indexed flash code (ILIFC)[6] seems promising because it is simple
and scalable in nature. The key idea of ILIFC is, the authors consider, the use
of slices which is a small group of flash cells. Through clever encoding rule, a
slice is able to store both of the index and the value of a bit of the data. This
mechanism assigns more number of cells to more frequently written bits, and
4 Flash Code with Dual Modes of Encoding
contributes to reduce the write deficiency. The problem with ILIFC is that the
size of a slice is fixed to k, which requires that kpnand some degradation of
write deficiency. To overcome this problem, the authors have studied encoding
techniques such as the binary-indexed slice encoding and its e↵ective uses[8][9].
The flash code which is discussed in the following sections can be regarded as
the extension of those studies of the authors.
3 Dual-Mode Flash Code
In this study, we propose a new flash code which we refer to as dual-mode flash
code (DMFC). This flash code incorporates two modes of encoding; stacked
segment encoding (SS encoding) and binary-indexed slice encoding (BS encod-
ing)[9]. SS encoding works fine if all data bits experience almost the same num-
bers of write operations, but its efficiency is degraded if some data bits are writ-
ten more frequently than others. To get around this problem of SS encoding, we
consider to temporally switch to BS encoding, and accommodate excess write
operations for those frequently written bits. The use of BS encoding brings a
certain overhead, but the BS encoding is able to incorporate the “non-uniform”
situation in which some bits are written more frequently than others. In the
following, we first introduce the two encoding schemes which are used in the
proposed flash code.
3.1 Stacked Segment Encoding
A simple encoding rule, which we call stacked segment encoding (SS encoding), is
introduced in this section. A segment is a group of kcells in an erase block, and
astack of segments is an ordered collection of segments. Let hbe the number of
segments contained in a stack, and let Si=(ci,1,...,c
i,k1)with0ih1
denote the i-th segment in the stack. In this encoding, the value ci,j is managed
in such a way that ci,j is not empty only if all of c0,j ,...,c
i1,j are full. In
asense,hcells c0,j ,...,c
h1,j are stacked over, and used from the bottom of
the stack. Those hcells form a single virtual cell whose value can be h(q1)
at the maximum, and represents the value djof the j-th bit of the data by
dj=(
Ph1
i=0 ci,j ) mod 2. The encoding operation is obvious; to flip the value of
dj, we determine the smallest integer isuch that ci,j is not full, and increase
the value of ci,j by one. If all of c0,j,...,c
h1,j are full, then we allocate a new
segment, push that new segment on the top of the existing stack, and raise the
value of ch,j in the newly stacked segment. A block erasure is called if we cannot
allocate a new segment in the erase block.
This SS encoding works fine if all data bits experience almost the same num-
ber of write operations. However, we need to retain many active segments, which
results in large write deficiency, if there is big di↵erence among the numbers of
write operations for data bits. To mitigate this issue, we include another mech-
anism for recording excess write operations, and invoke that mechanism if we
already have too many active segments. To implement a trigger mechanism for
Flash Code with Dual Modes of Encoding 5
that switch, a slightly modified encoding function is defined and denoted by ES,m
with man integer parameter. The function ES,m works almost the same way as
the above explanation, but ES,m refuses to allocate a new segment and issues a
failure signal if the stack already has mactive segments. We use DS,m to denote
the decoding function of this encoding.
3.2 Binary-Indexed Slice Encoding
The binary-indexed slice encoding (BS encoding) is an encoding scheme which is
similar to (the first-phase of) ILIFC but makes use of smaller slices than ILIFC.
Aslice is a set of scells where s=kin ILIFC, and sis the smallest even
integer which satisfies sblog2(k+ 1)c+ 1 in this BS encoding. An active slice
S=(c0,...,c
s1)representsabitindexi(S) and a bit value v(S)byitself.The
value djof the data is regarded as one if and only if the erase block contains an
active slice Swith i(S)=jand v(S) mod 2 = 1. In the BS encoding, slices are
operated through the “binary-indexing” rule which is reviewed in this section.
See [9] for more detailed explanation of the rule. We also assume that readers
are familiar with the principle of the index-less indexed encoding in [6].
The binary-indexing encoding rule consists of four phases. The phase-1 rule
is used only when an empty slice S=(c0,...,c
s1) is “activated”, that is, the
slice is assigned an index j. In this phase, cell values are set so that (c0···cs1)2
becomes the binary representation of j+ 1 (i.e., j+1=c02s1+···+cs120).
Cells having the value of 1 after this operation are called type-1 cells, and cells
having the value of 0 are called type-0 cells. Note that a slice contains both of
type-0 and type-1 cells because 1 j+1 k, and therefore the Hamming weight
of (c0···cs1)2is more than 0 and less than s. Further encoding operations are
defined so that type-0 cells and type-1 cells are always distinguishable, and we
can find that i(S)=jby identifying the types of cells.
The phase-2 rule is applied as far as we have a type-1 cell which is not full. In
this phase, the encoding function selects a type-1 cell that has the lowest value
among all type-1 cells, and raise the value of the selected type-1 cell by one.
The types of cells are distinguishable because all type-0 cells are empty while all
type-1 cells are not.
The phase-3 rule is applied if the criteria of the phase-2 does not hold and
there is at least one type-0 cell whose value is less than q2. The rule in this
phase-3 is similar to that of the phase-2, but the encoding function selects a
type-0 cell that has the lowest value, and raise the value of the selected type-0
cell by one. Note that the cell cannot be full because the value of the selected
type-0 cell was less than q2. Therefore, all type-0 cells remain not full, while
all type-1 cells are full.
The final phase-4 rule is used if the values of cells in the slice are either of
q2 (for the type-0 cells) or q1 (for the type-1 cells). In this phase, all cells
are filled up and the slice is brought to full.
Figure 1 illustrates the change of the state for the case of s= 4, q= 4 and
j= 4. In this case, j+ 1 = 5 = (0101)2and the second and the fourth cells are
type-1. Type-1 cells are operated in the phase-2, and type-0 cells are operated
6 Flash Code with Dual Modes of Encoding
0 0 0 0 0 1 0 1 0 2 0 1 0 2 0 2 0 3 0 2 0 3 0 3 1 3 0 3 1 3 1 3 2 3 1 3 2 3 2 3 3 3 3 3
phase-1 phase-2 phase-3 phase-4
Fig. 1. The operation of cell values in binary-indexing rule
in the phase-3. Remark that type-0 cells are never filled-up in the phase-3, and
the type of cells in an active slice are always distinguishable.
The bit value v(S) is defined as the number of operations that the slice has
experienced so far, that is, v(S)=(
Ps
i=0 ciwj+ 1) mod 2, where wjis the
Hamming weight of the binary representation of i(S)+1. We denote the encoding
and decoding functions of BS encoding by EBand DB, respectively.
3.3 Dual-Mode Flash Code
To accommodate both of the SS encoding and BS encoding in one erase block
(c0,...,c
n1), we consider to allocate segments of SS encoding from the begin-
ning of the erase block, and slices of BS encoding from the end of the erase block.
The allocation is done in an adaptive manner. If ES,m requests the i-th segment,
then the segment is allocated at (cik ,...,c
(i+1)k1). If EBrequests the i-th slice,
then the slice is allocated at (cn1(i+1)s+1 ,...,c
n1is). In both cases, the al-
location fails if the erase block does not contain enough number of empty cells.
To prevent possible collision of the allocation of segments and slices, we consider
to keep sor more cells empty between the region of segments and the region
of slices. By scanning slices from the end of the block, we can precisely identify
the region of slices. The region of segments are then easily identified because the
two regions cannot overlap.
The encoding function Eof the dual-mode flash code (DMFC) is described
by utilizing the encoding functions ES,m and EB. Given a bit index jand a
current state C,E(j, C) first invokes ES,m(j, C ) and tries to record the write
operation by the SS encoding. If ES,m (j, C) successfully records the change, the
encoding completes. If, unfortunately, ES,m(j, C ) issues the failure signal, then
E(j, C) invokes EB(j, C) and tries to record the write operation by BS encoding.
In case EB(j, C) cannot accommodate the write operation, E(j, C ) returns E
and requests a block erasure. The decoding function Dis simply defined as
D(C)=(DS,m (C)+DB(C)) mod 2.
3
2
1
04
c0c1
5
3
1
c2
9
6
2
c3c4
segment 0
c5c6
10
8
7
c7
15
13
11
c8c9
segment 1
empty
cells
c92
16
c93 c94 c95
slice 1
c96 c97
14
12
c98
17
12
c99
slice 0
Fig. 2. A demonstration of the encoding process of DMFC
Flash Code with Dual Modes of Encoding 7
Figure 2 illustrates how the cell values are raised in the encoding of DMFC.
We assume that n= 100, k= 5 and q= 4. In this which means that only
two segments can be simultaneously active. Each column in the simultaneously
setting, a slice consists of s= 4 cells. We also assume that m= 2 array of boxes
corresponds to a cell, and boxes represents the use of cell levels. If 17 write
operations are performed on bits
2,3,2,0,2,3,2,2,3,2,3,2,3,2,3,3,2
in this order, then cell levels are used as illustrated in the figure. Up to the 11th
write operation (counting from 1), the writes are recorded in segments by using
the SS encoding. The 12th write on the bit 2 is not accommodated by the SS
encoding because it requires a third active segment which is not allowed, and we
switch to the BS encoding. A slice 0 is allocated, and the third and the fourth
cells in the slice become type-1 because 2 + 1 = (0011)2. Meanwhile, the writes
on the bit 2 is accommodated by the BS encoding, but it has the chance to
switch back to the SS encoding when more write operations are performed and
the segment 0 becomes full.
4 Results and Analysis
4.1 Four Factors of Write Deficiency
The write deficiency of DMFC is contributed by four factors. The first and the
second factors are the write deficiencies made by active segments and active
slices, which we denote by 1and 2, respectively. The third factor 3of the
write deficiency is the contribution made by full slices.3It is shown in [9] that
one full slice contributes s2 write deficiency. If we can determine the number
✓of full slices, then 3=✓(s2). The fourth factor 4is made by cells which
do not belong to any segments or slices. Using the number of such cells, we
can write 4=(q1).
4.2 Worst Case Write Deficiency
Depending on the parameters n,k,qand m, there are two di↵erent scenarios
which result in the largest write deficiency.
The first scenario is such that write operations are made only on one bit until
the block erasure occurs. In this case, the SS encoding accommodates the first
m(q1) write operations, and the other write operations are all made through
the BS encoding using binary-indexed slices. At the time of block erasure, we
have mactive segments which occupy mk cells, no active slice, ✓=b(nmk)/sc
1 full slices, and =nmk ✓scells which do not belong to any segments
or slices. Each active segment contains one full cell and k1 empty cells, and
3Di↵erent from slices, full segments makes no contribution to the write deficiency,
and we can ignore full segments in the analysis of write deficiency.
8 Flash Code with Dual Modes of Encoding
therefore one such segment contributes (k1)(q1) write deficiency each. We
have 1=m(k1)(q1), and the write deficiency in this scenario is given by
1+2+3+4=m(k1)(q1) + 0 + ✓(s2) + (q1).(2)
In the second scenario, m(q1) + 1 write operations are made to k1 data
bits, and then, write operations are made to the remaining one bit until the block
erasure occurs. This pattern of write operations results in mfull segments, m
active segments, k1 active slices, ✓=b(n2mk (k1)s)/sc1fullslices
and =n2mk (k1+✓)scells which do not belong to any segments
or slices. Similar to the previous scenario, each active segment contributes (k
1)(q1) write deficiency. The active slice is used only once, and each active slice
contributes s(q1) 1 write deficiency. Consequently,
1+2+3+4=(k1)((m+s)(q1) 1) + ✓(s2) + (q1).(3)
There is no particular order between the values of (2) and (3); the former
can be larger than the latter for some specific parameters, and the converse is
true for some other parameters. In an asymptotic “big-O” notation, however,
(3) gives the worse (larger) write deficiency because (2) is in O(qk +n) and (3)
is in O(qk log k+n).
The write deficiency of DMFC contains O(n) factor, and therefore worse
than ILIFC. However, as we see in the next section, DMFC shows remarkably
superior write deficiency in the average case.
4.3 Average Case Write Deficiency
Computer simulation was performed in order to estimate the average case write
deficiency of DMFC against ILIFC and some other flash codes. It should be
noted that ILIFC in its original form [6] consists of two main phases, but we only
implemented the first phase of ILIFC since, as observed in [3], the contribution
of the second phase is not significant in a non-asymptotic discussion. For the
simulations, we fixed the values of n= 2048 and q= 8, and investigated the
results at di↵erent values of k. For each kvalue, 30 experiments were run, and
the average write deficiency percentage was gathered. It is assumed that all k
data bits have the equal probability to be selected by write operations.
Figure 3 shows the average write deficiency percentage of several flash codes
including ILIFC and DMFC. DMFC-4 and DMFC-6 stand for DMFC with m=
4 and m= 6, respectively. To understand the contribution of the dual-mode
encoding, the graph also shows the performance of “single-mode” flash codes; SS-
only (simulated) shows the write deficiency of a flash code that makes use of the
SS encoding technique only (the number of active segments is not bounded in this
case), and BS-only shows the write deficiency of a flash code that makes use of
the BS encoding technique only. We can see that DMFC shows the smallest write
deficiency among all flash codes compared. Unlike ILIFC and BS-only encoding,
DMFC and SS-only encoding do not have a degeneration point where the write
Flash Code with Dual Modes of Encoding 9
!
!"#
!"$
!"%
!"&
'
'"#
(#!($!(%!(&!('!!('#!('$!('%!('&!(#!!(
!"!#$ %&'()*+
&&'()*+,-./01*23456 78#$'9
78#$': &&'()*+,-2)2*+3/;2*6
Fig. 3. Write deficiency percentage.
deficiency percentage shoots up close to 1. This favorable characteristic is due to
the use of segments which allow us to start encoding by assigning only one cell to
each bit. This is not possible in a pure slice-based flash code, and DMFC inherits
the advantage of the segment-based encoding. We can also see that DMFC shows
better performance than SS-only encoding. This is because DMFC makes use of
slices whose size is much smaller than segments. Consider for example that, after
several encoding operations, we have k1 empty cells in an erase block. The
space is too small to accommodate a new segment, but sufficient to accommodate
several slices. In this sense, DMFC inherits the advantage of the slice-based
encoding. The e↵ect of the parameter mis not clear in these experiments. For
small values of k, we can see that DMFC-6 shows better performance than
DMFC-4. For large values of k, the di↵erence of mmakes little sense because,
if kis large, then an erase block is able to accommodate a limited number of
segments.
We have seen that the write deficiency of DMFC is contributed by four fac-
tors. Figure 4 shows the contribution of each factor on the write deficiency. The
graph shows that the contribution of 3and 4are almost negligible. The ma-
jor factors of write deficiency are 1and 2which are made by active segments
and active slices, respectively. Figure 4 shows that the value of 2changes reg-
ularly according to the value of k. This phenomenon can be explained in terms
of the number of slices. Except for very small values of k, the DMFC tends to
allocate as many segments as possible in the erase block. This means that only
nmod kcells are used to accommodate slices, and the number of slices is about
(nmod k)/s. These slices are once assigned, but not used so much. Almost all
10 Flash Code with Dual Modes of Encoding
!
!"#
!"$
!"%
!"&
!"'
!"(
!")
!"*
'$!'&!'(!'*!'#!!'#$!'#&!'#(!'#*!'$!!'
!"#$%& '(
') '*
'+
Fig. 4. An analysis of the di↵erent factor of write defiency of DMFC
cell levels in the slices are left unused, which implies that 2is approximated as
2=(s(q1) 1) (nmod k)
s.(4)
Even though this discussion is quite harsh, the formula well explains the exper-
imental result in Fig. 4.
For the estimation of 1, we consider the number of segments which are used
in SS-only encoding. Even though DMFC and SS-only encoding are di↵erent,
the estimation result on the SS-only encoding helps in analyzing DMFC also.
In Section 3.1, we saw that cells in a stack of segments constitute a virtual cell,
and we have kvirtual cells in a stack of segments. Let Xt
wwith w0 and
t0 be a random variable denoting the number of virtual cells whose values
are wor more after twrite operations. Obviously X0
0=kand X0
w= 0 for
w>0. With one write operation, one of virtual cells is selected and its value is
increased by one. If a cell with value wis selected at the t-th write operation,
then Xt+1
w+1 =Xt
w+1 + 1. This event occurs with probability of (Xt
wXt
w+1)/k,
and therefore the expected values of Xt
wsatisfy
E[Xt+1
w+1]=E[Xt
w+1]+(E[Xt
w]E[Xt
w+1])/k. (5)
If we regard that the h-th segment is allocated when E[Xt
(h1)(q1)+1]1, then
(5) defines a certain relation between tand h. Figure 5 shows the growth of
the number of segments obtained from (5), and that gathered from computer
simulations for DMFC-6 and SS-only encoding. We can see that the recursive
relation (5) accounts for the number of segments used in DMFC. Let hmax =
bn/kcbe the maximum number of segments which can be allocated in an erase
block, and let tmax be the smallest integer such that E[Xtmax
hmax(q1)+1 ]1. The
value of tmax can be regarded as the estimation of the number of write operations
accommodated by SS-only encoding. Unfortunately tmax is not derived by a
closed-form formula, but we can determine tmax by a computer search. The
curve of SS-only (analytical) in Fig. 3 shows the value of tmax for each value of
k. The analytical values are slightly larger than the values gather by computer
Flash Code with Dual Modes of Encoding 11
!"#$%&$'(#
!
"
#!
#"
$!
$"
%!
#$!#&!#'!#(!##!!##$!##&!##'!##(!#$!!#$$!#$&!#$'!#
!!"#$%&'($)
**+,
!!"#$%&'($)-.#/01%21'3
Fig. 5. The growth of the number of segments
simulation, but give good approximation of the average write deficiency of SS-
only encoding. Using the value of tmax,vw=E[Xtmax
w]E[Xtmax
w+1 ] can be
regarded as the expected number of virtual cells whose values are w. A virtual
cell with value wcontributes m(q1) wwrite deficiency, and the value of 1,
which is the write deficiency contributed by active segments, is given as
1=m(q1)km(q1)(E[Xtmax
m(q1)])
m(q1)1
X
w=0
vww. (6)
Because 3and 4are negligible, the average write deficiency is approximated
as 1+2, which is computed by using (4) and (6). We have clarified all four
factors of the write deficiency of DMFC, though some computer search is needed
to determine the value of 1.
5 Conclusion
In this paper, a new flash code, called dual-mode flash code (DMFC), is proposed.
DMFC inherits advantages of the stacked segment encoding and binary-indexed
slice encoding, and shows remarkable write deficiency in the average case. The
paper also investigates the average write deficiency of DMFC, and proposed an
analytical estimation of the write deficiency. Further studies can focus on provid-
ing a more rigorous discussion of the average performance, and the derivation of
a closed-form formula of the write deficiency. Furthemore this paper introduces
a new method for constructing a flash code. Further research can also involve
investigating combining di↵erent modes of encoding to create new flash codes.
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12 Flash Code with Dual Modes of Encoding
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