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International Journal of Multimedia and Ubiquitous Engineering
Vol.9, No.9 (2014), pp.381-396
http://dx.doi.org/10.14257/ijmue.2014.9.9.38
ISSN: 1975-0080 IJMUE
Copyright ⓒ 2014 SERSC
The K-Partition Flash Code with BIFC-based Sharing and some
Variants
Riz Rupert L. Ortiz and Proceso L. Fernandez, Jr.
Ateneo de Manila University
rizortiz@gmail.com, pfernandez@ateneo.edu
Abstract
Flash codes are used to handle decoding and encoding of digital information to flash
memory devices. The performance of a flash code is usually evaluated using the write
deficiency metric. This paper introduces the K-Partition Flash Code (KPFC) with BIFC-
based sharing and explores some of its variants KPFC is a coding scheme that involves a
sharing mechanism within partitions of a flash memory block. The technique was designed to
allow more cell writes to flash devices in order to improve its performance by lowering its
write deficiency. Computer simulations were conducted to estimate the average case
performances of the flash codes. Simulation results showed that its performance is generally
better than the flash codes in literature.
Keywords: block erasure, block-write, flash code, flash memory
1. Introduction
Flash memory is a non-volatile semi-conductor storage device. Flash memory has a wide
variety of applications nowadays. It has become increasingly important that it is even utilized
significantly in some embedded systems used by intelligent applications such as household
appliances, telecommunication devices and other high technology machinery. It has become a
dominant nonvolatile memory because it is cheap, fast, and reliable. Moreover, flash memory
can be electrically programmed and erased with relative ease [1].
Flash memory has a hierarchical structure organized into blocks, where a block can contain
thousands of memory cells. The memory cell is the smallest unit for performing read and
write operations to flash memory. Each cell, which has multiple possible states, can store
electric charges using some rules provided from a coding scheme that can also retrieve the
data from the values stored in the flash memory. This coding mechanism is referred to as
floating codes or commonly known now as flash codes [2, 9].
The process of adding a charge in a cell is called cell programming [4] which is typically
done through electron injection [7]. Using the encoding function from some coding scheme,
the level of charge of a cell can be increased. In multilevel flash memories, there can be two
possible problems that may occur when performing cell writes: errors when too many
electrons are added (overshoots) or errors when too few electrons are added (undershoot).
There is difficulty in lowering a charge to flash memories. Thus, overshoots are more of a
problem than undershoots [6].
Decreasing the charge of a cell is very costly and should be avoided as much as possible.
While it is easy to raise the level of charge in a cell, to lower it is difficult and time
consuming. This property of flash memory is referred to as write asymmetry [2]. In fact,
lowering a charge in a single cell only is not allowed. Such lowering of charges can only be
made through a block erasure, the process that erases the charges of all the cells within a
International Journal of Multimedia and Ubiquitous Engineering
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382 Copyright ⓒ 2014 SERSC
memory block. Unfortunately, flash memory is constrained by a limited number of erase
cycles. Currently, typical flash memory can only allow up to about 105 block erasures [2],
although there are already some flash devices that are capable of 106 block erasures [5]. The
block erasure process is slow and, after some repetitions, causes the memory chip to wear out.
Thus, block erasures significantly reduce the longevity, reliability and speed of flash
memories [7]. Continued block erasures will make memory blocks unreliable for storing data
and will eventually damage the flash memory when the block erasures exceed the allowed
number of erase cycles.
One way of extending the lifespan of flash memory is to improve the hardware technology
to allow for more block erasures. Another approach is to design flash codes that are efficient
so that more cell writes can be made to flash memories before calling a block erasure. The
focus of this study is to come up with efficient coding schemes that can increase the lifespan
of flash memories. Such coding schemes can delay the occurrence of block erasures and,
ultimately, prolong the lifespan of flash memory. The most salient point is to maximize the
number of write operations or bit updates and delay the occurrence of a block erasure.
2. Preliminaries
Flash memory is composed of a number of blocks. A block in particular is comprised of
thousand of cells, the exact number of which is denoted by n in this paper. The typical values
of n are between 218 and 222 [8]. In multilevel flash memories, each cell can be in one of q
levels from a finite set Aq = {0,1,…, q-1}. Thus, a block can be abstractly represented as
vector C = (c0, c1,…, cn-1), where ci {0,1,…, q-1}. The parameter q ranges from 2 to 256
[8]. Values of every cell can be increased or decreased by injecting a charge through the
process called the Fowler-Nordheim tunneling mechanism or hot-electron injection
mechanism [7], where electrons are trapped and in return determine the threshold voltage of
the cell. The number of trapped electrons concentrates around q discrete levels that
correspond to the q cell states [2]. A cell with a charge of 0 is empty, while a cell with a
charge of q-1 is full. A cell that is neither empty nor full is said to be active.
The information vector encoded in a flash memory block, is a k-bit data D = (d0, d1, ...,
dk−1), where di {0,1} and k < n. A block using binary cells normally stores 64, 128, or 256
kilobytes of data [2, 3, 10].
The mechanism to store the k-bit data D to the block state vector C is managed by some
flash code F as implemented in the flash memory. Formally, the flash code F = (E, D) is a
coding scheme consisting of two main functions. The encoding function E (i,C) provides the
rules on writing a new state to the block given the index i of the data bit di that needs to be
updated and the current state C of the block. The decoding function D (C) interprets the
content C of the block into the corresponding k-bit data D [2].
Flash code operates on both the cell and the block levels. The metric normally used to
evaluate the performance of the flash code is its write deficiency. It is defined as
(1)
where n(q − 1) is the theoretical maximum number of cell writes allowed for a block of n
cells with q levels each, while t is the actual number of write operations that the flash code F
is able to perform before calling a block erasure [11]. Alternatively, the write deficiency ratio
to compare flash codes can be expressed as
International Journal of Multimedia and Ubiquitous Engineering
Vol.9, No.9 (2014)
Copyright ⓒ 2014 SERSC 383
(2)
The range of possible values is [0,1], with
as the worst performance and
as
the ideal one.
3. Related Literature
Flash codes are recently attracting the attention of researchers, especially in the area of
computer science. More studies are coming out because of the possibilities that can be made
in developing and improving new and efficient coding schemes. This section provides
background on some flash codes in literature.
3.1 Index-less Indexed Flash Code and Layered Index-less Indexed Flash Code
The index-less Indexed Flash Code (ILIFC) and Layered Index-less Indexed Flash Code
(LILIFC) are two of the popular flash codes in literature. Both coding schemes operate on
sub-blocks called slices. A block of n cells is partitioned into equally sized slices with k cells
each. ILIFC and LILIFC offer an elegant scheme of storing both the bit index and bit value
from the active slices. As to performance, while both flash codes have the same worst case
write deficiency, LILIFC performs better than ILIFC in the average case. The asymptotic
worst case write deficiency of both flash codes is O(k2q). Refer to [8,11] for more detailed
discussions on how these two flash codes manage the decoding and encoding of data to flash
memory.
3.2 Bimodal Flash Code and 2-Split Bimodal Flash Code
The Bimodal Flash Code (BMFC) of Esling et al., uses two modes of encoding [12, 13].
Instead of having k cells per slice, the flash code reduced the slice size to k/2, effectively
lowering the write deficiency and improving its performance. Results of the study showed
that BMFC returned a better average case performance than ILIFC and LILIFC. A variant
called 2 Split BMFC (BMFC2) further reduced the size of slice to k/4, which also resulted to
a better average case performance than BMFC. The worst case write deficiency of BMFC and
BMFC2 is O(k2q + n/k). Both flash codes showed that performance can be improved by
utilizing a reduced size of slices.
3.3 Binary Indexed Flash Code and BIFC-RCM
Tan and Kaji introduced the Binary Indexed Flash Code (BIFC) [14]. Similar to BMFC,
the Binary Indexed Flash Code uses fewer cells for each slice. Specifically, the slice has a
size s = O(log k), where s is the smallest even number satisfying s ≥ 1+log2(k+1). When k is
sufficiently large, results of the study revealed that BIFC has a better write deficiency than
ILIFC, LILIFC, BMFC and BMFC2 [12-14]. However, there is an overhead deficiency of s-2
for every slice, because of the implicit indexing. To mitigate this problem, another flash code
was proposed by Tan and Kaji to address this overhead issue. It is called the BIFC with
Resizable Cluster Method (BIFC-RCM) [15, 16]. BIFC-RCM uses the BIFC technique that
capitalizes on smaller slice size and incorporates a growing slice to mitigate the effect of the
overhead problem. The BIFC and BIFC-RCM have asymptotic worst case write deficiencies
of O(qk log k + n) and O(qk log k + rk log k log((n/k) log k)), respectively.
International Journal of Multimedia and Ubiquitous Engineering
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384 Copyright ⓒ 2014 SERSC
3.4 Simple Segmentation Flash Code and the Phoenix Flash Code
In Simple Segmentation Flash Code (SSFC), a block is divided into equally-sized sub-
blocks, called segments. The block C = (c0, c1,..., cn−1) has exactly n/k segments. The ith
segment (0 ≤ i < n/k) corresponds to the vector Si = (cik, cik+1, ... , cik+k−1). Within each
segment, the jth entry (0 ≤ j < k) is assigned to the jth bit. SSFC at its best case has a very
good performance when each of the k bits has uniform or equal frequencies for write
operations. On the other hand, Phoenix Flash Code (PFC) is a novel scheme that allows the
process of absorption and revival operations. Similar to SSFC, the erase block C = (c0, c1,...,
cn−1) has exactly n/k segments, where the jth element of the segment corresponds to the jth
bit. The idea of introducing absorption stems from the fact the SSFC has a very high write
deficiency in the worst case. SSFC has an asymptotic worst case write deficiency of O(nq)
[17,18], while PFC has O(k2q + n) [19].
4. Computer Simulation for Average Case Write Deficiency
This section describes the various procedures involved in the completion of this study.
Figure 1 illustrates how the study is organized and subsequently conducted. Basically, it
involves three phases:
1. Benchmarking of some Flash Codes in Literature - The first phase refers to the
implementation of some flash codes in literature to provide benchmark results
relative to their performance. Write deficiency ratios were returned by the computer
simulations and were later used for comparison. Flash codes with promising
performance were taken into consideration to gain some insights for the design and
development of the proposed flash codes.
2. Design and Development of the Proposed Flash Codes - After careful studies on the
strengths and weaknesses of some flash codes in literature, some key concepts were
integrated to the proposed flash codes. The developed flash codes were thoroughly
tested to ascertain the correctness of the implementation.
3. Performance Analysis - In this study, computer simulation was used to estimate the
average case performance of the flash codes. Computer simulations were
implemented in java (see Table 1 for the parameters used in the simulations).
Empirical results for the average case performances of the flash code were returned
by the simulations, which are set to run in 30 experiments with fixed parameters of
n=2048 and q=8. The experiments were performed for various k values from 4 to
n/2, in increments of 4.
Benchmarking of
some Flash Codes
in Literature
Design and
Development of the
Proposed Flash Codes
Performance Analysis
(Average Case Write
Deficiency)
Figure 1. The Methodology of the Study
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Copyright ⓒ 2014 SERSC 385
Table 1. Parameters used in the Computer Simulations
Variable
Values
Description
n
2048
Size of the Block
k
4,8,12, …, 1024
Information Vector Length
q
8
Number of States
E
30
Number of Experiments
The computer simulation used two types of distributions to examine the performance of
the proposed flash codes. The following are the sample frequency distributions:
1. Uniform Frequency Distribution – Each of the k-bit data has an equal chance of being
selected. For instance, if the simulation uses 4-bit data, then each bit has a 1/k (25%)
probability of being selected for bit update. Consequently, all the bits have
approximately equal total number of bit writes at the point of block erasure,
2. Steady Dominated Distribution – In this type of distribution, a certain bit is given a
higher probability for bit update, i.e., that bit dominates the write operation in the
computer simulation. For simplicity, this study examines the performance of several
flash codes using 50% and 70% steady dominated distributions only. It simply means
that a bit is set to 0.5 or 0.7 probability for bit update (see Table 2 for a 4-bit data
illustration), while the remaining probability is distributed uniformly to the remaining
bits.
Table 2. Steady Dominated Distribution
4-bit Data
50% Dominated
70% Dominated
Bit 0
0.500
0.7
Bit 1
0.167
0.1
Bit 2
0.167
0.1
Bit 2
0.167
0.1
5. K-Partition Flash Code with BIFC-based Sharing and its Variants
This section describes the K-Partition Flash Code with BIFC-based Sharing and its
variants, namely KPFC-s, KPFC-r and KPFC-m. The performances of the flash code and its
variants are shown for both uniform and steady dominated distributions.
The basic K-Partition Flash Code (KPFC) partitions the block of n cells into k groups of
contiguous cells, with each partition having exactly n/k cells. The k partitions are mapped
to the k bits of data. When a bit update is performed on the ith bit, the level of charge of a cell
within the corresponding ith partition is increased.
For the purpose of illustration, a sample write sequence is shown in Figure 2. In this
example, the data vector D is (1,1,1,1,) after the fourth bit update; each of the partitions has
one charge added. Note that a block erasure only occurred when the partition for bit 0 is
already full and the current bit index to be updated is also 0 (refer to the 11th bit update).
Using computer simulation, the average case performance of KPFC was estimated, and a
summary of the write deficiency across different k values is shown in Figure 3. Regardless of
the availability of other partitions to accommodate cell write, a block erasure is still inevitable
using this coding scheme since each of the k bits is only assigned to one and distinct partition.
With this scenario, an improvement to KPFC is proposed using some sharing mechanism.
International Journal of Multimedia and Ubiquitous Engineering
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386 Copyright ⓒ 2014 SERSC
5.1. KPFC-s
The first variant of KPFC with sharing mechanism is referred to as KPFC-s. In this flash
code, the block is still divided into partitions to represent each of the k bits from the data
vector. Technically, each partition also has n/k cells. To avoid the scenario from KPFC that
triggers the block erasure when a bit update is to be done on a full partition, a sharing
mechanism is introduced. The technique utilizes a BIFC slice, which has a slice size of s ≥
1+log2(k+1) (a detailed discussion on how each BIFC slice is initialized is available in [14]).
Note that the BIFC slice must satisfy the constraint s < n/k so that a BIFC slice can be
integrated to a normal partition. Hence, instead of calling for a block erasure when a write
operation is to be performed to an already full partition, the KPFC-s encoding looks for
another partition where there is enough contiguous cells to accommodate a BIFC slice and
perform the necessary bit update. See Fig. 4 for the illustration of a sample write sequence
for KPFC-s.
To avoid ambiguity in decoding, a separator empty cell is maintained within a partition
with a BIFC slice. A BIFC slice can be extended one cell at a time if there is a need for more
cells, a scenario that occurs when there is a dominating bit index to be encoded. Extending the
Figure 3. The Performance of KPFC using Computer Simulation Results.
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, … , 400})
Figure 2. A Sample Write Sequence for KPFC
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BIFC slice is allowed if there are still available cells within the partition. Since a separator
cell can also contribute to write deficiency, it is discarded when the BIFC slice needs to take
over the whole partition assuming the original bit assigned to the partition has no bit update at
all. Please refer to Figure 5 on how a BIFC slice is extended with KPFC-s. In effect, a block
erasure occurs if an attempt to initialize a BIFC slice fails.
The implementation of a sharing mechanism is only possible when s < n/k. However, if s
exceeds the number of cells assigned to every partition, a BIFC slice cannot be made
available within a normal partition. Thus, the flash code is reduced to the original KPFC.
Using empirical results, the write deficiency of KPFC is slightly reduced using the sharing
mechanism of KPFC-s. Note that depending on the number of cells of a block, denoted by n,
and the length of the data vector, denoted by k, the expected improvement of KPFC-s is only
Figure 4. A Sample Write Sequence for KPFC-s
Figure 5. Extending a BIFC Slice in KPFC-s
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388 Copyright ⓒ 2014 SERSC
possible when s < n/k. When s ≥ n/k, the performance of KPFC-s is exactly the same as
that of KPFC (see Figure 6).
The present setup of KPFC-s assigns exactly n/k for each partition and does not utilize
the remainder (n mod k) cells. These remainder cells can be at most k-1. When k is large, the
remainder cells can significantly affect the write deficiency of the flash code. Technically,
KPFC-s just leaves these remainder cells untouched. It is with this context that KPFC-s can
still be enhanced by applying some mechanism to utilize those unassigned cells to allow more
cell writes and possibly reduce the write deficiency.
5.2. KPFC-r
KPFC-r is another variant of KPFC that introduces a method to utilize those unassigned
cells not used by KPFC and KPFC-s. Instead of being left discarded, a new technique is
introduced to KPFC so that these remainder cells can still be used. There are two scenarios
Figure 6. The Performance of KPFC and KPFC-s
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, …, 400})
Figure 7. KPFC-r Slice Allocation
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where KPFC-r allocates and utilizes the unassigned cells to the existing partitions of the
block. When s ≤ n/k, the flash code automatically assigns the cells not part of any partition
to the last partition, i.e., partition k-1 has a bigger size compared to the other partitions. For
simplicity of implementation, the last slice absorbs those extra cells so that a sharing BIFC
slice can be initialized and extended if possible. If r = n mod k refers to the number of
unassigned cells, the size of slice k-1 in this scheme is n/k+ r. For example, if n=30 and k=8,
then r=6, therefore, slice k-1 will have a size of 9 cells while the rest will have a size of only 3
cells each (See Figure 7 Case 1). This strategy works well when k is not sufficiently large.
The second scenario happens when s > n/k, in which case it is no longer possible to
initialize a sharing BIFC slice. Instead of reverting to the original KPFC where a simple
mapping is implemented, KPFC-r may still use up some of the unassigned cells. Because r =
n mod k, then at most k-1 cells can be distributed and therefore at most k-1 partitions can have
one added cell each (See Fig. 7 Case 2). On the average, half of the partitions can have an
extra cell in this KPFC variant, thus potentially lowering the write deficiency further. The
performance of the flash code is improved because of the technique where unassigned cells
are still utilized as much as possible. Refer to Fig.8 for a comparison of simulation results for
KPFC, KPFC-s and KPFC-r.
5.3. KPFC-m
The last variant of KPFC is called KPFC-m. The idea behind KPFC-m is to initialize as
many BIFC slices as possible within a normal partition. Unlike KPFC-s that can only allocate
a single BIFC slice within a partition, multiple BIFC slices can be accommodated in KPFC-m
as long as there are available contiguous cells for sharing mechanism (see Fig.9). Similar to
KPFC-r, the unassigned cells are also utilized in KPFC-m. However, the remainder cells are
automatically assigned to the last partition. Hence, the last partition will have more cells
compared to the rest of the partition. Contrary to KPFC-s and KPFC-r, extending the BIFC
slice in KPFC-m is not possible. If a bit update is mapped to an already full BIFC slice, the
Figure 8. The Performance of KPFC, KPFC-s and KPFC-r
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, …, 400})
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390 Copyright ⓒ 2014 SERSC
flash code looks for adjacent empty cells that can accommodate a BIFC slice. Block erasure
is returned if search for vacant cells returns a null.
With the introduction of multiple BIFC slices within normal partition, it turns out that the
flash code returned the best write deficiency among the variants of KPFC. With more BIFC
slices made, more write operations were performed (see Table 3). Consequently, call for
block erasures were delayed KPFC-m performs better when there is uniform frequency of
distribution. As shown in Figure 10, KPFC-m outperforms KPFC, KPFC-s and KPFC-r.
KPFC-m is even better than KPFC-s when s ≤ n/k (Note that KPFC-s has the ability to
extend its BIFC slice by a single cell only). It can also be inferred from the graph that it has
a superior mechanism to utilize the unassigned cells compared to KPFC-r, most especially for
large values of k. KPFC-m significantly reduces the write deficiency of the flash code
resulting to a better performance.
Table 3. Comparison between KPFC-m and KPFC-r in terms of
Average Write Operations
(n=2048, q=8, k {4, 8, … , 1024}, E=30 experiments)
k
KPFC-r
KPFC-m
Additional Writes
4
14254
14287
33
204
10582
10649
68
404
8645
9505
860
604
6709
8753
2044
804
4991
7541
2550
1004
5581
6793
1212
Figure 9. Initializing BIFC-slices in KPFC-m
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5.4. KPFC-r and KPFC-m in Steady Dominated Distributions
To examine further the performance of the proposed flash codes, the variants with better
write deficiency in uniform distribution were tested on steady dominated distributions. As
discussed in the previous section, a dominating bit index is designed in this type of
distribution. The performances of the KPFC-r and KPFC-m in 50% and 70% steady
dominated distributions were returned after running the simulation. Results are shown in
Figure 11a and Figure 11b, respectively.
Given the graph of KPFC-r and KPFC-m, the results show that the latter is generally better
than the former for both 50% and 70% steady dominated distributions. However, the write
deficiency of both flash codes eventually increases as the value of k grows. Note that KPFC-r
degenerates at k>200, because the write deficiency ratio shoots up close to 1 already. On the
contrary, KPFC-m can still manage to lower it due to its inherent capability of utilizing the
extra cells of the block. Initializing multiple BIFC slices potentially uses these unassigned
cells.
Figure 10. The performance of KPFC, KPFC-s, KPFC-r and KPFC-m
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, …, 400})
Figure 11.a KPFC-r and KPFC-m in 50% Steady Dominated Distribution
(n=2048, q=8, k {4, 8, … , 400})
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6. Comparison of KPFC variants with existing Flash Codes
This section presents the derived average write deficiency of KPFC, KPFC-s, KPFC-r and
KPFC-m. Moreover, the KPFC-r and KPFC-m were compared to the flash codes in literature.
Figure 12 shows the write deficiency ratios of KPFC, KPFC-s, KPFC-r and KPFC-m for
larger range of values for k, unlike in the previous figures that were only up to k=400. As
depicted in the graph, KPFC-s has a lower write deficiency for some lower values of k when
compared to the original KPFC. Basically, this is due to the sharing mechanism employed in
KPFC-s. Note that KPFC and KPFC-s converge starting at k when s > n/k. Nonetheless,
another variant, KPFC-r, performs better than the two former flash codes for all instances of
k. This is due to the utilization of the remainder cells in KPFC-r. In general, it is the KPFC-m
that has the best performance among all the variants of KPFC. Apparently, KPFC-m returns
the lowest write deficiency ratios for all values of k. In using multiple BIFC slices, KPFC-m
allows more cell writes, and this results in a delay to the occurrence of block erasures.
KPFC-r and KPFC-m are then compared to the existing flash codes in literature. The
performance of the flash codes as compared to slice based flash codes like ILIFC, LILIFC,
BMFC and BMFC2 are shown in Figure 13, where both KPFC-r and KPFC-m showed a
remarkable performance. When compared to more recent flash codes like BIFC and its
improved variant BIFC-RCM, the proposed flash codes still returned the better average case
performance (see Figure 14). KPFC-r and KPFC-m are then compared to high performing
flash codes like SSFC and PFC. Simulation results reveal that the proposed flash codes still
outperform the SSFC and PFC, as shown in Figure 15. With the introduction of sharing
mechanisms, the write deficiency of the flash codes is significantly reduced, contributing to
the better performance of the flash codes in the average case.
Figure 11. b KPFC-r and KPFC-m in 70%, Steady Dominated Distribution
(n=2048, q=8, k {4, 8, … , 400})
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Figure 12. Comparing KPFC, KPFC-s, KPFC-r and KPFC-m
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, … , 1024})
Figure 13. Comparing ILIFC/LILIFC, BMFC, BMFC2,KPFC-r and KPFC-m,
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, … , 1024})
Figure 14. Comparing BIFC, BIFC-RCM, KPFC-r and KPFC-m
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, … , 1024})
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394 Copyright ⓒ 2014 SERSC
7. Conclusion
The K-Partition Flash Code with sharing mechanism is presented in this study.
There were three variants that were explored: KPFC-s, KPFC-r and KPFC-m. The
average case performances of the flash codes were estimated through computer
simulations. Overall, KPFC-r and KPFC-m returned the best performance. Empirical
results showed that the aforementioned variants of KPFC are more efficient than the
existing flash codes in literature. Further analysis shows that the write deficiency due
to unassigned cells has a significant effect on the over-all performance of the flash
codes. Hence, careful handling of these remainder cells is very valuable and essential
in lowering the write deficiency of flash codes, as shown in KPFC-r and KPFC-m.
Further studies can be made to explore new methods of utilizing cells in a block to
maximize cell writes. It is also interesting to conduct investigations on coming up
with novel and promising ways of encoding such as new framework and hybrid flash
codes, among others.
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Figure 15. Comparing SSFC, PFC, KPFC-r and KPFC-m
(Uniform Frequency Distribution, n=2048, q=8, k {4, 8, … , 1024})
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Vol.9, No.9 (2014)
Copyright ⓒ 2014 SERSC 395
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Authors
Riz Rupert L. Ortz, is a Ph.D. in Computer Science student at the
Ateneo de Manila University. He is currently connected to the
College of Management and Information Technology (CMIT) at
Northwest Samar State University (NwSSU), Calbayog City, Samar,
Philippines.
Proceso L. Fernandez, Jr., Ph.D., is an Associate Professor at
the Ateneo de Manila University.
International Journal of Multimedia and Ubiquitous Engineering
Vol.9, No.9 (2014)
396 Copyright ⓒ 2014 SERSC
