A Deliberate Bit Flipping Coding Scheme for Data-Dependent Two-Dimensional Channels
Summary (3 min read)
Introduction
- In principle, the ultimate coding approach for such datadependent channels is to design a set of sufficiently distinct error correction codewords that also satisfy channel constraints [15], [16].
- The theory of 1-D constrained coding is mature as well as practical aspects of 1-D code and decoder design.
- (a) (b) Fig. Paper Organization: Section II presents the notations and definitions used throughout the paper.
- In Section IV, the problem of minimizing the number of flipped bits in the DBF method is formulated.
II. NOTATIONS AND DEFINITIONS
- The authors denote a discrete random variable with an upper case letter (e.g., X) and its realization by the lower case letter (e.g., x).
- Throughout the paper, white squares denote zero bits and black squares represent 1. Consider a k-ominoe P and the set of all 2k binary configurations of that shape XP .
III. CHANNEL MODEL
- The authors call this set of P-shaped configurations the set of harmful configurations.
- Fig. 4. 2-D isolated-bits patterns containing the bit xi,j .
- The authors use the concept of polyominoes to just demonstrate the effect of harmful configurations on its neighboring bits over a 2-D binary pattern, also known as Remark 1.
- The probability that the channel is in the bad state (or, in the good state) depends on the input probability distribution, also known as Remark 3.
- The authors assume that the set of harmful patterns for the channel is the set of 2-D isolated-bits patterns, which are given in Fig.
IV. PROBLEM FORMULATION
- The authors want to send the pattern x over the communication channel in Section III, with the list of harmful configurations XBPi,j .
- The theory of constrained coding began with Claude Shannon’s classical 1948 paper [11], “A Mathematical Theory of Communications.”, also known as Remark 4.
- Finding the error pattern which removes a given set of 2-D configurations from a 2-D pattern and has the minimum Hamming weight via an exhaustive search among all admissible error patterns can be computationally prohibitive for large patterns, also known as Remark 5.
V. A PROBABILISTIC GRAPHICAL FORMULTION FOR MINIMZING BIT FLIPS
- The authors devise a probabilistic graphical formulation for the problem of minimizing the number of bit flips in the DBF method.
- In the following, the authors present a probabilistic formulation using a graphical model to find approximate solution for this problem using the GBP algorithm.
- For each bit xi,j ∈ Am,n, the distortion now is defined as the probability of having a distorted pattern xPi,j which has the Hamming distance wH(x̂Pi,j⊕xPi,j ) with x̂Pi,j 6∈ XBPi,j .
- In [39] and [47], it is shown that the region-based approximation (RBA) method provides an approximate solution for the partition function by minimizing the region-based free energy (as an approximation to the variational free energy).
- The authors first define a factor graph representation for the problem (maximizing p (x̂|x) in (30) for a given input pattern x subject to the constraint that x̂ ∈ S) and then formulate the RBA scheme for finding an approximate solution for this constrained maximization problem.
VI. NUMERICAL RESULTS
- The authors present numerical analyses of the GBPbased DBF method for removing harmful patterns.
- Without loss of generality, the authors focus on the 2-D isolated-bits configurations in all their experiments.
- The authors first present the analysis on statistics of the number of flipped bits for removing 2-D isolated-bits patterns from random 2-D patterns.
- To illustrate the usefulness of DBF method, the authors investigate its performance over the data-dependent channel in Section III under different scenarios in terms of the probability of uncorrectable bit errors, where the harmful configurations for the channel are the 2-D isolated-bits patterns.
- Finally, the authors compare the performance of the DBF method on a memoryless BSC with the row-by-row and bit-stuffing constrained coding schemes for the 2-D n.i.b. constraint, presented in [40] and [10] respectively.
A. Statistics of The Number of Bit Flips for Removing 2-D Isolated-Bits Patterns
- The performance of the DBF method relies on the error correction capability of the code being used, and of course the number of deliberate bit errors.
- Therefore, it is necessary to find how many bits in average are flipped within a codeword, and how this number compares to the error correction capability of the code.
- The authors have extracted the statistics of the number of bit flips for removing 2-D isolated-bits patterns from random 2-D patterns by the DBF method.
- Using the flipping probabilities in Fig. 8 and (32), the UBER is calculated for BCH codes of length 1024 with different rates (and consequently dmin).
- The choice of λ in the probabilistic formulation of problem, (28), depends on the constraint and the underlying method for solving the minimization problem.
B. Performance Evaluation of The GBP-Guided DBF Method
- The authors investigate the usefulness of DBF method for data-dependent 2-D channels, where specific patterns in channel inputs are the main cause of errors.
- For different values of αb and αg , the authors compare the average probability of error with and without incorporating the DBF method.
- Prior to transmission over the channel, the 2-D isolatedbits patterns are removed from the input pattern by flipping minimum number of bits.
- The transmitted pattern and channel output without DBF are x(m) and x(m)⊕ êCH, respectively.
C. Comparison Results on BSC
- The authors compare the proposed scheme of imposing the 2-D n.i.b. constraint by deliberate errors against the row-by-row and the bit-stuffing coding schemes on a BSC.
- The encoder first generates two sequences with different statistics, Bernoulli(1/2) and Bernoulli(1/3), from the sequence of information bits using a probability transformer.
- The redundancy for imposing the constraint is now used in their scheme to strengthen the ECC (BCH code), resulting in a gain over the other schemes.
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"A Deliberate Bit Flipping Coding Sc..." refers background in this paper
...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]....
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61 citations
"A Deliberate Bit Flipping Coding Sc..." refers background or methods in this paper
...Constraint [10]: The bit-stuffing method for mapping binary random sequences into a 2-D rectangular array satisfying the 2-D n....
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...constraint, presented in [40] and [10] respectively....
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...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]....
[...]
...As another example, in two-dimensional magnetic recording channels, 2-D isolated-bits patterns [10] are shown empirically...
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59 citations
"A Deliberate Bit Flipping Coding Sc..." refers background or methods in this paper
...We only consider these four patterns out of 32 possible patterns by BCH-[15, 5, 7] code as they cover all different flipping scenarios using the deliberate error insertion method....
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...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]....
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...In the following, we provide examples of BCH-[15, 5, 7] codewords that are arranged into 3 × 5 arrays, as they help to explain the concepts we have introduced so far....
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...For the above systematic BCH-[15, 5, 7] code (where the codewords are arranged into 3 × 5 arrays and the first row is equipped with the user bits), we identified the minimum number of bit flips required for removing 2-D isolated bit patterns from each of the possible BCH-[15, 5, 7] codewords....
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...Example 2: We assume that the user messages are the following binary vectors of length 5, m1 = (0, 1, 0, 0, 0), m2 = (1, 0, 0, 0, 0), m3 = (0, 1, 1, 1, 1) and m4 = (0, 1, 1, 0, 1), and are encoded by the triple-error correcting BCH[15, 5, 7] code....
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59 citations
50 citations
"A Deliberate Bit Flipping Coding Sc..." refers methods in this paper
...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]....
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