Classifications

• • G—PHYSICS
  • G11—INFORMATION STORAGE
  • G11C—STATIC STORES
  • G11C16/00—Erasable programmable read-only memories
  • G11C16/02—Erasable programmable read-only memories electrically programmable
  • G11C16/06—Auxiliary circuits, e.g. for writing into memory
  • G11C16/34—Determination of programming status, e.g. threshold voltage, overprogramming or underprogramming, retention
  • G11C16/3418—Disturbance prevention or evaluation; Refreshing of disturbed memory data
  • G11C16/3431—Circuits or methods to detect disturbed nonvolatile memory cells, e.g. which still read as programmed but with threshold less than the program verify threshold or read as erased but with threshold greater than the erase verify threshold, and to reverse the disturbance via a refreshing programming or erasing step
• • G—PHYSICS
  • G11—INFORMATION STORAGE
  • G11C—STATIC STORES
  • G11C11/00—Digital stores characterised by the use of particular electric or magnetic storage elements; Storage elements therefor
  • G11C11/56—Digital stores characterised by the use of particular electric or magnetic storage elements; Storage elements therefor using storage elements with more than two stable states represented by steps, e.g. of voltage, current, phase, frequency
  • G11C11/5621—Digital stores characterised by the use of particular electric or magnetic storage elements; Storage elements therefor using storage elements with more than two stable states represented by steps, e.g. of voltage, current, phase, frequency using charge storage in a floating gate
  • G11C11/5642—Sensing or reading circuits; Data output circuits
• • G—PHYSICS
  • G11—INFORMATION STORAGE
  • G11C—STATIC STORES
  • G11C16/00—Erasable programmable read-only memories
  • G11C16/02—Erasable programmable read-only memories electrically programmable
  • G11C16/06—Auxiliary circuits, e.g. for writing into memory
  • G11C16/26—Sensing or reading circuits; Data output circuits
• • G—PHYSICS
  • G11—INFORMATION STORAGE
  • G11C—STATIC STORES
  • G11C7/00—Arrangements for writing information into, or reading information out from, a digital store
  • G11C7/02—Arrangements for writing information into, or reading information out from, a digital store with means for avoiding parasitic signals

Definitions

• determining and/or calculating inter-cell interference models may require solving complex problems.
• determining two-dimensional inter-cell interference in a flash memory device may require solution of an NP-hard problem.
• use of some inter-cell interference models may be unrealistic and/or may not be useful.
• the present disclosure generally describes concepts and technologies for linear programming based decoding for memory devices.
• data is obtained at a computing device configured to execute a controller for managing writing data to and reading data from a data storage device such as a flash memory device, a phase change memory device, and/or other types of volatile and/or nonvolatile memory devices (hereinafter referred to as "memory devices").
• a data storage device such as a flash memory device, a phase change memory device, and/or other types of volatile and/or nonvolatile memory devices (hereinafter referred to as "memory devices").
• memory devices volatile and/or nonvolatile memory devices
• the controller and/or the computing device can be configured to read data from the memory device, taking into account expected voltage shifts and/or voltage migrations.
• the computing device also can be configured to identify a cell to be read, detect a cell threshold value level for the cell, and identify interference cells.
• the computing device also can determine interference voltage levels of the interference cells and decode the cell threshold voltage level of the read cell based upon a set of beliefs determined and/or known for the memory device.
• the computing device can be configured to determine the set of beliefs for a memory device by modeling the inter-cell interference between the cell and the interference cells as an integer programming problem.
• the computing device can relax the integer programming problem to obtain a linear program, and solve the linear program to obtain a set of beliefs that minimizes the linear program.
• the computing device can be configured to decode the cell threshold voltage level in accordance with the set of beliefs.
• a method for determining a value of a memory cell of a flash memory device can include detecting a cell threshold voltage level of the memory cell, determining an interference voltage level of an interference cell that interferes with the memory cell, and decoding the cell threshold voltage level in accordance with a set of beliefs to determine the value of the memory cell.
• the set of beliefs can include a minimization of an objective function of a linear program representing inter-cell interference between the memory cell and the interference cell.
• the set of beliefs can include a set of marginal probabilities satisfying a constraint associated with the linear program.
• the method also can include receiving a model representing inter-cell interferences associated with two or more cells of the flash memory device.
• the model can include the set of beliefs.
• the set of beliefs can be obtained by modeling the inter-cell interference between the memory cell and the interference cell as an integer programming problem, relaxing the integer programming problem to obtain the linear program, and solving the linear program to obtain the set of beliefs.
• the method also can include identifying, for two or more signals written to two or more cells of the flash memory device, resulting threshold voltage values, and determining, based upon the resulting threshold values, a configuration of writing to the two or more cells.
• the configuration can be a configuration that minimizes a Euclidean distance between the resulting threshold voltage values of the two or more cells and measured threshold voltage levels of the two or more cells.
• the method also can include determining an integer programming problem that represents the Euclidean distance between the resulting threshold voltage values and measured threshold voltage levels of the two or more cells, relaxing the integer programming problem to obtain a linear programming problem, and identifying the set of beliefs.
• the set of beliefs can correspond to a set of marginal probabilities that minimizes the linear programming problem.
• Detecting the cell threshold voltage level can include detecting the cell threshold voltage level in response to receiving a request to read the value of the memory cell.
• Decoding the cell threshold voltage level in accordance with the set of beliefs to determine the value of the memory cell can include compensating for the inter-cell interference between the memory cell and the interference cell by multiplying the cell threshold voltage level by a marginal probability corresponding to at least one of the set of beliefs, and determining the value of the memory cell according to a result of multiplying the cell threshold voltage level by the marginal probability.
• decoding the cell threshold voltage level in accordance with the set of beliefs to determine the value of the memory cell can include compensating for the inter-cell interference between the memory cell and the interference cell by multiplying the cell threshold voltage level by a marginal probability corresponding to at least one of the set of beliefs, determining a probability that the value of the memory cell is equal to zero, and determining a further probability that the value of the memory cell is equal to one. The probability can be compared to the further probability. In response to a determination that the probability is greater than the further probability, the value of the memory cell can be determined to be zero. In response to a determination that the further probability is greater than the probability, the value of the memory cell can be determined to be one.
• a computer readable medium can include computer executable instructions that, when executed by a computer, cause the computer to detect a cell threshold voltage level of a memory cell of a flash memory device, determine an interference voltage level of an interference cell that interferes with the memory cell, and decode the cell threshold voltage level in accordance with a set of beliefs to determine a value of the memory cell.
• the set of beliefs can include a minimization of an objective function of a linear program representing inter-cell interference between the memory cell and the interference cell.
• the computer readable medium further includes computer executable instructions that, when executed by the computer, cause the computer to receive a model that represents inter-cell interferences between two or more cells of the flash memory device.
• the model can include the set of beliefs.
• the set of beliefs can include a set of marginal probabilities that satisfies a constraint associated with the linear program.
• the computer executable instructions when executed by the computer, further cause the computer to model the inter- cell interference between the memory cell and the interference cell as an integer programming problem, relax the integer programming problem to obtain the linear program, and solve the linear program to obtain the set of beliefs.
• the computer readable medium further includes computer executable instructions that, when executed by the computer, further cause the computer to identify, for two or more signals written to two or more cells of the flash memory device, resulting threshold voltage values and measured threshold voltage levels, and determine, based upon the resulting threshold voltage values, a configuration of writing to the two or more cells.
• the configuration can minimize a Euclidean distance between the the resulting threshold voltage values and the measured threshold voltage levels.
• the computer readable medium further can include computer executable instructions that, when executed by the computer, further cause the computer to determine an integer programming problem that represents the Euclidean distance between the resulting threshold values and the measured threshold voltage levels, relax the integer programming problem to obtain a linear programming problem, and identify the set of beliefs.
• the set of beliefs can correspond to a set of marginal probabilities that minimizes the linear programming problem.
• the computer executable instructions when executed by the computer, can further cause the computer to detect the cell threshold voltage level in response to a request to read the value of the memory cell.
• the computer executable instructions when executed by the computer, can further cause the computer to multiply the cell threshold voltage level by a marginal probability corresponding to at least one of the set of beliefs to compensate for the inter-cell interference between the memory cell and the interference cell, and determine the value of the memory cell according to a result of multiplying the cell threshold voltage level by the marginal probability.
• the computer executable instructions when executed by the computer, can further cause the computer to multiply the cell threshold voltage level by a marginal probability corresponding to at least one of the set of beliefs to compensate for the inter-cell interference between the memory cell and the interference cell, determine a probability that the value of the memory cell is equal to zero, and determine a further probability that the value of the memory cell is equal to one.
• the probability can be compared to the further probability.
• the value of the memory cell In response to a determination that the probability is greater than the further probability, the value of the memory cell can be determined to be zero.
• the value of the memory cell can be determined to be one.
• a computing device can include a flash memory device including two or more cells, and a processor coupled to the flash memory device.
• the processor can be configured to execute computer executable instructions to receive a request to read a value of a memory cell of the flash memory device, and obtain a mathematical model that represents inter-cell interferences of the two or more cells.
• the model can include a set of marginal probabilities that satisfies constraints associated with the mathematical model.
• the processor also can be configured to execute the computer executable instructions to detect a cell threshold voltage level of the memory cell, determine, based upon the mathematical model, an interference voltage level of an interference cell that interferes with the memory cell, and decode the cell threshold voltage level in accordance with the set of marginal probabilities to determine the value of the memory cell.
• the set of marginal probabilities can include a minimization of an objective function of a linear program representing the inter-cell interferences.
• the processor can be further configured to execute the computer executable instructions to generate the mathematical model as an integer programming problem, relax the integer programming problem to obtain the linear program, and solve the linear program to obtain the set of marginal probabilities that minimize the linear program.
• the processor can be further configured to execute the computer executable instructions to identify, for two or more signals written to the two or more cells of the flash memory device, resulting threshold voltage values and measured threshold voltage levels, and determine, based upon the resulting threshold values, a configuration of writing to the two or more cells. The configuration can minimize a Euclidean distance between the resulting threshold voltage values and the measured threshold voltage levels.
• the processor can be further configured to generate the mathematical model based, at least partially, upon the Euclidean distance.
• the processor can be further configured to execute the computer executable instructions to determine an integer programming problem that represents the Euclidean distance between the resulting threshold values and the measured threshold voltage levels, relax the integer programming problem to obtain a linear programming problem, and identify the set of marginal probabilities.
• the set of marginal probabilities can correspond to a set of marginal probabilities that minimizes the linear programming problem.
• FIGURE 1 is a block diagram illustrating an operating environment for various embodiments of the concepts and technologies disclosed herein for linear programming based decoding for memory devices
• FIGURE 2 is a line diagram illustrating an example memory device for various embodiments of the concepts and technologies disclosed herein for linear programming based decoding for memory devices;
• FIGURE 3 is a flow diagram illustrating an example process for determining a value of a memory cell
• FIGURE 4 is a flow diagram illustrating an example process for determining a set of beliefs for use in determining a value of a memory cell
• FIGURE 5 is a line diagram illustrating additional aspects of the concepts and technologies disclosed herein for linear programming based decoding for memory devices
• FIGURE 6 is a block diagram illustrating an example computer capable of providing linear programming based decoding for memory devices.
• FIGURE 7 is a schematic diagram illustrating computer program products for linear programming based decoding for memory devices
• a computing device, controller, or other hardware and/or software can be configured to read data from a cell of a memory that may experience interference from one or more interference cells.
• One or more interference cells associated with the read cell can be identified, and threshold voltage levels associated with the read cell and the one or more interference cells can be determined.
• Inter-cell interference between the one or more interference cells and the read cell can be determined, modeled, and used to decode the threshold voltage level read from the read cell.
• the inter-cell interference can be modeled as an integer programming problem.
• the integer programming problem can be relaxed to obtain a linear programming problem, and the linear programming problem can be solved to obtain a set of beliefs and/or probabilities that can minimize or otherwise can optimize the linear programming problem.
• the identified set of beliefs can be used to decode the threshold voltage level of the read cell.
• FIGURE 1 is a block diagram illustrating an operating environment 100 for various embodiments of the concepts and technologies disclosed herein for linear programming based decoding for memory devices, arranged in accordance with at least some embodiments presented herein.
• the operating environment 100 shown in FIGURE 1 includes a computing device 102.
• the computing device 102 operates as a part of and/or in communication with a communications network ("network") 104, though this is not necessarily the case.
• the functionality of the computing device 102 can be provided by a personal computer (“PC”) such as a desktop computer, a tablet computer, and/or a laptop computer.
• PC personal computer
• the functionality of the computing device 102 can be provided by other types of computing systems including, but not limited to, server computers, handheld computers, netbook computers, embedded computer systems, personal digital assistants, mobile telephones, smartphones, other computing devices, combinations thereof, or the like.
• the computing device 102 can be configured to execute an operating system
• the operating system 106 can include a computer program for controlling the operation of the computing device 102.
• the application programs can include various executable programs configured to execute on top of the operating system 106 to provide functionality associated with the computing device 102.
• the application programs can include programs for providing various functions associated with the computing device 102 such as, for example, multimedia application programs, web browser application programs, messaging application programs, productivity application programs, and/or other application programs. It should be understood that these examples of application programs are illustrative, and should not be construed as being limiting in any way.
• the computing device 102 also can include a data storage device (hereinafter referred to as a "memory") 108.
• the functionality of the memory 108 can be provided by one or more volatile and/or non-volatile memory devices including, but not limited to, a flash memory device, a phase-change memory device, a hard disk drive, and/or other memory storage devices.
• the functionality of the memory 108 can be provided by a flash memory device. Because the functionality of the memory 108 can be provided by additional and/or alternative types of data storage devices, it should be understood that this embodiment is illustrative, and should not be construed as being limiting in any way.
• the memory 108 can be configured as a removable and/or non-removable memory device. As such, while the memory 108 is illustrated as a component of the computing device 102, it should be understood that the memory 108 can be removed from the computing device 102 at various times and/or for various reasons.
• the computing device 102 can be configured to store data 110 in the memory
• the computing device 102 can be configured to store the data 110 in the memory 108 by altering voltages of one or more memory cells (hereinafter referred to as "cells") 112 of the memory 108.
• the memory 108 can include hundreds, thousands, or even millions of cells 112.
• the cells 112 can be arranged in a grid structure that includes one or more rows and/or one or more columns of the cells 112.
• FIGURE 2 One example embodiment of the memory 108 and cells 112 thereof is illustrated and described below with reference to FIGURE 2. Briefly, a row of the cells 112 may correspond to a word line, and a column of the cells 112 may correspond to a bit line, though this is not necessarily the case.
• a memory 108 can include hundreds, thousands, or even millions of word lines and/or bit lines.
• the computing device 102 can be configured to include, to execute, and/or to access a controller 114 that can be configured to manage various processes described herein for reading and/or decoding the data 110 stored in the memory 108.
• the controller 114 can be provided by a hardware controller.
• the controller 114 can be provided by a software application or module executed by the computing device 102.
• the controller 114 can be provided as a part of the operating system 106.
• the controller 114 is described herein as a software module executed by the computing device 102 as part of the operating system 106 or as a separate application program. In light of the above variations described above, it should be understood that this example is illustrative, and should not be construed as being limiting in any way. The functionality of the controller 114 is described in further detail below.
• the computing device 102 can be configured to access and/or receive the data 110 from various sources.
• the data 110 can be generated by the computing device 102 and stored in the memory 108.
• the data 110 can be obtained from a data source 116 that can be configured to operate as a part of and/or in communication with the network 104.
• the functionality of the data source 116 can be provided by a network hard drive, a server computer, a data store, and/or other real or virtual devices.
• the computing device 102 can be configured to write the data 110 to the memory 108.
• the computing device 102 also can be configured to execute the controller 114 to read and/or decode the data 110 stored in the memory 108.
• the computing device 102 can be configured to execute the controller 114 to read and decode the data 110 stored in the memory 108.
• the controller 114 can be configured to identify a cell 112 of the memory 108 that is to be read. The selection of the cell 112 by the controller 114 can be based upon various considerations. For example, the controller 114 can select the cell 112 to be read based upon a read request received by the computing device 102 and/or the controller 114. Additionally, or alternatively, the controller 114 can be configured to select the cell 112 to be read based upon known properties of the memory 108. Because the controller 114 can select the cell 112 in additional and/or alternative ways, it should be understood that these examples are illustrative, and should not be construed as being limiting in any way. The controller 114 can read a cell threshold voltage level of the selected cell 112.
• the controller 114 also can identify an interference cell 112 associated with the selected cell 112.
• the controller 114 can be configured to identify interference cells 112 for a particular selected or read cell 112 based upon known properties of the memory 108.
• the controller 114 can store and/or access data that indicates a known layout and/or properties of the memory 108 that can be used by the controller 114 to identify interference cells 112 for the selected cell.
• the controller 114 can identify the interference cell 112, and read an interference voltage level of the interference cell 112.
• the phrase "interference voltage level" of the interference cell 112 can be used herein to refer to a threshold voltage level of the interference cell 112.
• the threshold voltage level of the interference cell 112 can be a component or consideration in a function that can be used to decode a value of the read cell 112.
• the interference voltage level can correspond and/or be used to determine a known or expected effect on a voltage level of a read cell 112 from the interference cell 112.
• a read cell 112 is shown as a black cell with white text. As shown in FIGURE 1, the read cell 112 can experience a voltage level shift during a write process affecting a neighboring or nearby cell 112. This is illustrated in FIGURE 1 by showing that the read cell 112 can be affected by a voltage leaks or voltage migrations V m is (hereinafter collectively and/or generically referred to as "voltage migrations V m ”) from none, one, or more than one of the neighboring cells 112. While the read cell 112 is illustrated in FIGURE 1 as being affected by voltage migration V m from five neighboring cells 112, it should be understood a voltage shift at the read cell 112 can be caused by a single interference cell 112.
• each interference cell 112 may have an associated voltage migration V m that, compared to other voltage migrations V m of other interference cells 112, can be equal and/or can be unequal.
• any number of interference cells 112 may exist for a particular read cell 112. As such, the illustrated embodiment should be understood as being illustrative and should not be construed as being limiting in any way.
• the controller 114 can determine interference voltages associated with the one or more interference cells 112. Furthermore, the controller 114 can be configured to generate and/or apply a known or expected set of beliefs that can define a relationship between an interference voltage of an interference cell 112 and a value of a read cell 112. Thus, the controller 114 can be configured to determine an affect the interference cells 112 have on a read cell 112, thereby identifying the expected voltage shift experienced at the read cell 112 as a result of the interference cells 112.
• the value of a read cell 112 can correspond to a function of the threshold voltage level of the read cell 112 and an interference voltage of an interference cell 112. It should be understood that this example is illustrative, and should not be construed as being limiting in any way.
• the controller 114 can decode the detected cell threshold voltage level of the read cell 112. Thus, the controller 114 can identify the voltage shift and correct for the voltage shift to determine the value of the read cell. It should be understood that in some embodiments, the set of beliefs associated with a particular memory 108 may be known. In some other embodiments, the controller 114 can be configured to determine the set of beliefs, as is explained below.
• the controller 114 and/or the computing device 102 can be configured to obtain a set of beliefs for a particular memory 108 and/or array of cells 112.
• the set of beliefs can be used to decode cell threshold voltage levels of cells as mentioned above.
• the set of beliefs can be generated and/or known for a particular memory 108.
• the computing device can model the inter-cell interference for a particular memory 108 as an integer programming problem.
• the modeling can occur at various times. In some embodiments, the modeling can occur after receiving a read request.
• the computing device 102 can formulate an integer programming problem, relax the integer program problem into a linear program to minimize an objective function of the linear program and thereby obtain a set of beliefs, and decode the cell 112 being read according to the beliefs.
• FIGURE 1 illustrates one computing device 102, one memory 108, one controller 114, and one data source 116. It should be understood, however, that some implementations of the operating environment 100 include multiple computing devices 102, multiple memories 108, multiple controllers 114, and/or zero or multiple data sources 116. Thus, the illustrated embodiment of the operating environment 100 should be understood as being illustrative of one embodiment thereof and should not be construed as being limiting in any way.
• FIGURE 2 a line diagram illustrating an example memory device for various embodiments of the concepts and technologies disclosed herein for linear programming based decoding for memory devices, arranged according to at least some embodiments presented herein, will be described.
• FIGURE 2 illustrates an example memory device such as the memory 108 illustrated and described herein. It should be understood that the illustration of FIGURE 2 can correspond to only a portion of the memory 108 shown in FIGURE 1.
• the memory 108 can include word lines 200A-D (hereinafter collectively and/or generically referred to as "word lines 200"), which can correspond to the rows of the cells 112 mentioned above with reference to FIGURE 1. Additionally shown in FIGURE 2 are bit lines 202A-D (hereinafter collectively and/or generically referred to as "bit lines 202"). As shown in FIGURE 2, some embodiments of the memory 108 include an array of cells 112, wherein one or more of the cells 112 can include a control gate 204, a floating gate 206, a source 208, and a drain 210. According to some implementations of the memory 108, control gates 204 of cells 112 that lie in a same row can be connected to a same word line 200.
• sources 208 and drains 210 of cells 112 that lie in a same column can be connected to a same bit line 202. It should be understood that the cells 112 of the memory 108 shown in FIGURE 2 can correspond to one contemplated example of the cells 112 and therefore should not be construed as being limiting in any way.
• the computing device 102 can write the data 110 to the memory 108 by modifying threshold voltage levels of the cells 112.
• the computing device 102 can be configured to inject electrons into the floating gates 206 of the cells 112. Due to a parasitic capacitance-coupling effect that sometimes can be experienced within the memory 108, voltage levels of the control gates 206, sources 208, and drains 210 of neighboring cells 112 can be disturbed during writing of a cell 112, as explained above with reference to FIGURE 1. Thus, a particular cell 112 can experience a voltage shift from one or more neighboring cells 112.
• Voltage migration V m is shown in FIGURE 2, assuming that each cell 112 that borders the read cell 112, illustrated with dark shading in FIGURE 2, has been written. It should be understood that zero, one, or more than one cell 112 may be written, and therefore the illustration of FIGURE 2 is illustrative and should not be construed as being limiting in any way. Various aspects of identifying, modeling, and/or using inter-cell interference models to decode values of cells 112 will be illustrated and described in more detail below.
• FIGURE 3 a flow diagram illustrating an example process
• the process 300 is described as being performed by the computing device 102. It should be understood that this embodiment is illustrative, and should not be viewed as being limiting in any way. Furthermore, as explained above with reference to FIGURE 1, the computing device 102 can be configured to execute one or more applications, program modules, or other instructions including, but not limited to, the controller 114 to provide the functionality described herein.
• the process 300 may begin at block 302 (IDENTIFY A CELL TO BE
• the computing device 102 may be configured to identify a cell 112 that is to be read.
• the computing device 102 can identify the cell 112 that is to be read based upon a read request that specifies particular data and/or cells 112 to be read, based upon a configuration of the memory 108, and/or based upon additional and/or alternative considerations.
• the computing device 102 may be configured to receive a read request in block 302 and identify a cell 112 to be read based, at least partially, upon the read request.
• Block 302 may be followed by block 304.
• the computing device 102 may be configured to detect (e.g., measure) a cell threshold voltage level of the cell 112 identified and/or selected as described with reference to block 302. Because detecting and/or measuring a cell threshold voltage level of a cell 112 of a memory 108 is generally understood, additional details of detecting or measuring the cell threshold voltage level of the cell 112 will not be described in further detail with reference to block 304. Block 304 may be followed by block 306.
• the computing device 102 may be configured to identify one or more interference cells 112.
• the computing device 102 can be configured to identify the one or more interference cells 112 based upon a known structure and/or properties of the memory 108, based upon a known write order associated with the memory 108, other known and/or determined characteristics of the memory 108, combinations thereof, or the like.
• the functionality of the computing device 102 illustrated with respect to block 306 of the process 300 can correspond to determining, identifying, and/or obtaining data indicating one or more interference cells 112, though this is not necessarily the case.
• Block 306 may be followed by block 308.
• the computing device 102 may be configured to determine interference voltage levels of the interference cells 112.
• the phrase "interference voltage” can be used to refer to a threshold voltage level of a cell 112 identified as an interference cell 112.
• the functionality of the computing device 102 illustrated with respect to block 308 can correspond to the computing device 102 being configured to determine threshold voltage levels of the interference cells 112.
• the computing device 102 may be configured, via execution of the functionality illustrated in blocks 304-308, read the threshold voltage levels of the cell 112 identified in block 302 and the interference cells 112 identified in block 306. Block 308 may be followed by block 310.
• the computing device 102 may be configured to decode the value of the cell 112 identified by the computing device 102 in block 302.
• the computing device 102 can be configured to decode the cell threshold voltage level detected in block 304 based, at least partially, upon the one or more interference voltage levels determined in operation 308.
• the value of the read cell 112 can be a function of various data including, but not limited to, the threshold voltage level of the cell 112 and threshold voltage levels of interference cells 112.
• One example process for decoding the cell threshold voltage level based upon interference voltage levels is illustrated and described below with reference to FIGURE 4. Because the cell threshold voltage level can be decoded in additional and/or alternative manners, it should be understood that this example process is illustrative, and should not be construed as being limiting in any way.
• Block 310 may be followed by block 312.
• the computing device 102 can be configured to terminate execution of the process 300.
• the computing device 102 also may be configured to repeat (e.g., periodically, continuously, or on-demand) the process 300 by returning to block 302 from block 310, though this is not explicitly shown in FIGURE 3. As such, some embodiments of the process 300 can return to block 302 from block 312.
• the computing device 102 may be configured to terminate the process 300 at any time, as noted above.
• FIGURE 4 a flow diagram illustrating an example process
• the process 400 may begin at block 402 (MODEL THE INTER-CELL INTERFERENCE BETWEEN THE CELL AND THE INTERFERENCE CELLS AS AN INTEGER PROGRAMMING PROBLEM), wherein the computing device 102 may be configured to generate a model that models, simulates, or represents the inter-cell interference between the cell 112 that is being read, and one or more interference cells 112. It can be appreciated with reference to the process 300 described above that the model generated in block 402 can model inter-cell interference between the cell 112 identified in operation 302 and the one or more interference cells 112 identified in block 306. It should be understood that this example is illustrative, and should not be construed as being limiting in any way.
• the model generated in block 402 can correspond to an integer programming problem.
• This integer programming problem can represent an effect on a victim cell (the cell 112 being read) by one or more interference cells 112.
• the value of the read cell 112 can be related to a threshold voltage level of any number of interference cells 112.
• the summation included in the model generated in block 402 can represent interference from any number of interference cells 112 experienced by the read or victim cell 112.
• the computing system 102 can be configured to identify a configuration of X(i) such that the corresponding resulting threshold voltage levels Y(i) have a smallest Euclidean distance to the measured threshold voltage levels. If Z(i) is used to represent the measured threshold voltage level of an z ' -th memory cell 112, the square of the Euclidean distance may be expressed as ⁇ i ⁇ x(i) + ⁇ j h(i,j)X(J)— . Thus, if a block decoding approach to reducing reading errors is used, the result may be a configuration of X(i) such that the above Euclidean distance can be minimized.
• a signal written to a particular memory cell 112 may be a Boolean variable, e.g., a zero (0) or a one (1).
• the computing device 102 may be configured to determine that the square of X(i) may be equal to X(i).
• the computing device 102 may be configured to determine that the square of Z(i) also may be a constant number.
• the computing device 102 can be configured to reformulate the Euclidean distance expressed above as an integer programming problem.
• the computing device 102 can be configured to express the
• the computing device 102 also may be configured to determine that integer programming problems are NP-complete and to solve the above linear programming problem as a linear relaxation problem instead, as will be explained below with reference to block 404.
• Block 402 may be followed by block 404.
• the computing device 102 can be configured to relax the linear programming problem that represents the Euclidean distance to obtain a linear programming problem.
• the computing device 102 can relax the integer programming problem.
• the computing device 102 can be configured to reformulate the linear programming problem as minx(i) ⁇ i ⁇ j ⁇ x(ilx(j) b(i,j)X(QX(j)p(X(Q, X(j) ⁇ ) + ⁇ i ⁇ X( i) a ⁇ i)X ⁇ i)p ⁇ X ⁇ i)), subject to
• Block 404 may be followed by block 406.
• the computing device 102 can be configured to solve the linear programming problem developed in block 404 to identify a set of beliefs that solves the linear program.
• the computing device 102 can be configured to determine that the probabilities p(X(i)) and p(X(i),X(j)) identified in block 404 are a set of beliefs.
• a "set of beliefs" can include a set of marginal probabilities that satisfy certain marginal probability consistent constraints associated with the linear programming problem.
• the marginalization of p(X(i),X(j)) can be equal to p(X(i)), or the like.
• the computing device 102 can be configured to determine that the linear programming problem developed in block 404 can correspond to an optimization problem of the integer programming problem generated in block 402. Block 406 may be followed by block 408.
• the computing device 102 can be configured to decode a cell, for example the cell 112 identified in block 302 of the process 300, according to the set of beliefs obtained in block 406.
• the computing device 102 can be configured to determine a probability that the value of decoded cell 112 is equal to zero and determine a probability that the value of the decoded cell 112 is equal to one.
• the computing device 102 can compare these two probabilities to one another and determine the value of the cell 112 based upon the comparison. In particular, if the computing device 102 determines that the probability that the value of the decoded cell 112 is equal to one exceeds or is greater than probability that the value of the decoded cell 112 is equal to zero, the computing device 112 can be configured to determine that the value of the cell 112 is equal to one.
• the computing device 102 can be configured to terminate the process 400.
• the computing device 102 can be configured to output and/or store the set of beliefs instead of and/or in addition to decoding the cell threshold voltage level in accordance with the set of beliefs as described above with reference to block 408.
• the process 400 and/or other processes can include exporting or saving the set of beliefs in a memory or other data storage device.
• the set of beliefs for a particular memory 108 can be determined, used, and/or saved by the computing device 102.
• the computing device 102 can be configured to decode cell voltage levels of cells 112 without executing the process 400, though this is not necessarily the case.
• FIGURE 5 a line diagram illustrating additional aspects of the concepts and technologies disclosed herein for linear programming based decoding for memory devices, arranged according to at least some embodiments presented herein, will be described.
• FIGURE 5 illustrates an example array 500 of cells 112.
• the illustrated array 500 is provided for purposes of reference during description of an example of the concepts and technologies disclosed herein for linear programming based decoding. It should be clear that the example array 500 having nine cells 112 is illustrative, and that memory devices such as the memory 108 may have thousands, millions, or even billions of cells 112.
• Y(i,j) is used to refer to the threshold voltage level of a cell 112 located at an z ' -th column and a y ' -th row of the array 500.
• the notation X(i,j) is used herein to refer to a signal written to the cell 112 located at an z ' -th column and a y ' -th row of the array 500.
• the following example assumes interference from a cell 112 located at a neighboring upper row, and a cell 112 located at a left column.
• a cell located at the second row and second column (2,2) can be assumed, for purposes of this example, to experience interference from the cells 112 (1,2) and (2, 1) located in the first row and the first column, respectively. Because a particular read cell actually may experience interference from any number of neighboring cells 112, it should be understood that this example is illustrative, and should not be construed as being limiting in any way.
• values of the cells 112 of the array 500 can be expressed as:
• Y (l, 2) X(l, 2) + j x X(l, 1)
• Y (3, 2) X(3, 2) + x X X(2, 2) + y X X(2, 1)
• Y(l,3) X(l,3)+yxX(l,2)
• Y (2, 3) X(2, 3) + x X X(l, 3) + y X X(2, 2)
• Y (3, 3) X(3, 3) + x X X(2, 3) +y X X(3, 2), wherein x and y can be values measured for a particular memory 108 of which the array 500 is a part.
• x and y will be assumed to be 0.2 and 0.3, respectively. As such, the above values can be expressed as:
• Y (2, 2) X(2, 2) + 0.2 X X(1,2)H V 0.3 x X(2, 1)
• Y (3, 2) X(3,2) 4- 0.2 X X(2, 2) H V 0.3 x X(2, 1)
• Y (2, 3) X(2, 3) + 0.2xX(l,3)H V 0.3 x X(2, 2)
• Y (3, 3) X(3, 3) + 0.2 X X(2, 3) H V 0.3 x X(3, 2)
• the interference can be determined by measuring the voltage levels Y(i,j) set forth above and comparing those to actual observed measurement values Z(ij). For purposes of illustration, the following observed measurement values Z(ij) are assumed:
• Euclidean distance relationship set forth herein can be expressed as (Y (1, 1) - Z(l, l)) 2 + (Y (2, 1) - Z(2, l)) 2 + (Y (3, 1) - Z(3, l)) 2 + (Y (1, 2) - Z(l, 2)) 2 + (Y (2, 2) - Z(2, 2)) 2 + (Y (3, 2) - Z(3, 2)) 2 + (Y (1, 3) - Z(l, 3)) 2 + (Y (2, 3) - Z(2, 3)) 2 + (Y (3, 3) - Z(3, 3)) 2 .
• the Euclidean distance can be expressed as (X(l, 1) - Z(l, 1)) 2 + (X(2, 1) + 0.2X(1, 1) - Z(2, 1)) 2 + (X(3, 1) + 0.2X(2, 1) - Z(3, 1)) 2 + (X(l, 2) + 0.3X(1, 1) - Z(l, 2)) 2 + (X(2, 2) + 0.2X(1, 2) + 0.3X(2, 1) - Z(2, 2)) 2 + (X(3, 2) + 0.2X(2, 2) + 0.3X(2, 1) - Z(3, 2)) 2 + (X(l, 3) + 0.3X(1, 2) - Z(l, 3)) 2 + (X(2, 3) + 0.2X(1, 3) + 0.3X(2, 2) - Z(2, 3)) 2 + (X(3, 3) + 0.2X(1, 3) + 0.3X(2, 2) - Z(2, 3)) 2 + (X(3, 3) + 0.2X(1, 3) + 0.3X(2,
• the Euclidean distance can be expressed as 0.4X(1, 1)X(2, 1) + 0.4X(2, 1)X(3, 1) + 0.6X(1, 1)X(1, 2) + 0.4X(1, 2)X(2, 2) + 0.6X(2, 1)X(2, 2) + 0.12X(2, 1)X(1, 2) + 0.4X(2, 2)X(3, 2) + 0.6X(2, 1)X(3, 2) + 0.12X(2, 1)X(2, 2) + 0.6X(1, 2)X(1, 3) + 0.4X(1, 3)X(2, 3) + 0.6X(2, 2)X(2, 3) + 0.12X(2, 2)X(1, 3) + 0.4X(2, 3)X(3, 3) + 0.12X(3, 2)X(2, 3) - 3.51X(1, 1) - 2.63X(1, 2) - 2.2X(2, 1) - 1.29X(2, 2) - 2.8X(
• the expectation can be expressed as E[0.4X(1, 1)X(2, 1)] + E[0.4X(2, 1)X(3, 1)] + E[0.6X(1, 1)X(1, 2)] + E[0.4X(1, 2)X(2, 2)] + E[0.6X(2, 1)X(2, 2)] + E[0.12X(2, 1)X(1, 2)] + E[0.4X(2, 2)X(3, 2)] + E[0.6X(2, 1)X(3, 2)] + E[0.12X(2, 1)X(2, 2)] + E[0.6X(1, 2)X(1, 3)] + E[0.4X(1, 3)X(2, 3)] + E[0.6X(2, 2)X(2, 3)] + E[0.12X(2, 2)X(2, 3)] + E[0.12X(2, 2)X(1, 3)] + E[0.4X(2, 3)] + E[0.6X(2, 2)X(2, 3)] + E[0.12X(2, 2)X(1, 3)] + E[0.4X(2, 3)] + E[0.6X(2, 2)
• FIGURE 6 is a block diagram illustrating an example computer 600 capable of providing linear programming based decoding for memory devices arranged according to at least some embodiments presented herein.
• the computer 600 includes a processor 610, a memory 620 and one or more drives 630.
• the computer 600 may be implemented as a conventional computer system, an embedded control computer system, a laptop computer, a server computer, a set-top box (“STB") or set-top unit (“STU”), a gaming console, a vehicle information system, a mobile telephone or smartphone, other computing systems, other hardware platforms, combinations thereof, or the like.
• STB set-top box
• STU set-top unit
• the drives 630 and their associated computer storage media provide storage of computer readable instructions, data structures, program modules and other data for the computer 600.
• the drives 630 can include an operating system 640, application programs 650, program modules 660, and a database 680.
• the program modules 660 may include a controller, such as the controller 114.
• the controller 114 may be adapted to execute either or both of the processes 400 and/or 500 for linear programming based decoding for memory devices as described in greater detail above (e.g. , see previous description with respect to one or more of FIGURES 4 - 5).
• the computer 600 further includes user input devices 690 through which a user may enter commands and data.
• the input devices 690 can include one or more of an electronic digitizer, a microphone, a keyboard and pointing device, commonly referred to as a mouse, trackball or touch pad.
• Other input devices may include a joystick, game pad, satellite dish, scanner, other devices, or the like.
• These and other input devices can be coupled to the processor 610 through a user input interface that is coupled to a system bus, but may be coupled by other interface and bus structures, such as a parallel port, game port or a universal serial bus ("USB").
• Computers such as the computer 600 also may include other peripheral output devices such as speakers, printers, displays, the data source 116, and/or other devices, which may be coupled through an output peripheral interface 694 or the like.
• the computer 600 may operate in a networked environment using logical connections to one or more computers, such as a remote computer (not illustrated), the data source 116, and/or other devices operating as part of or in communication with a network 608 coupled to a network interface 696.
• the remote computer may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and can include many or all of the elements described above relative to the computer 600.
• Networking environments are commonplace in offices, enterprise-wide area networks ("WAN”), local area networks (“LAN”), intranets, and the Internet.
• the computer 600 When used in a LAN or WLAN networking environment, the computer 600 may be coupled to the LAN through the network interface 696 or an adapter. When used in a WAN networking environment, the computer 600 typically includes a modem or other means for establishing communications over the WAN, such as the Internet or the network 608.
• the WAN may include the Internet, the illustrated network 608, various other networks, or any combination thereof. It will be appreciated that other mechanisms of establishing a communications link, ring, mesh, bus, cloud, or network between the computers may be used.
• the computer 600 may be coupled to a networking environment.
• the computer 600 may include one or more instances of a physical computer-readable storage medium or media associated with the drives 630 or other storage devices.
• the system bus may enable the processor 610 to read code and/or data to/from the computer storage media.
• the media may represent an apparatus in the form of storage elements that are implemented using any suitable technology, including but not limited to semiconductors, magnetic materials, optical media, electrical storage, electrochemical storage, or any other such storage technology.
• the media may represent components associated with memory 620, whether characterized as RAM, ROM, flash, or other types of volatile or nonvolatile memory technology.
• the media may also represent secondary storage, whether implemented as the storage drives 630 or otherwise.
• Hard drive implementations may be characterized as solid state, or may include rotating media storing magnetically- encoded information.
• the storage media may include one or more program modules 660.
• the program modules 660 may include software instructions that, when loaded into the processor 610 and executed, transform a general-purpose computing system into a special-purpose computing system. As detailed throughout this description, the program modules 660 may provide various tools or techniques by which the computer 600 may participate within the overall systems or operating environments using the components, logic flows, and/or data structures discussed herein.
• the processor 610 may be constructed from any number of transistors or other circuit elements, which may individually or collectively assume any number of states. More specifically, the processor 610 may operate as a state machine or finite-state machine. Such a machine may be transformed to a second machine, or specific machine by loading executable instructions contained within the program modules 660. These computer- executable instructions may transform the processor 610 by specifying how the processor 610 transitions between states, thereby transforming the transistors or other circuit elements constituting the processor 610 from a first machine to a second machine.
• the states of either machine may also be transformed by receiving input from the one or more user input devices 690, the network interface 696, other peripherals, other interfaces, or one or more users or other actors. Either machine may also transform states, or various physical characteristics of various output devices such as printers, speakers, video displays, or otherwise.
• Encoding the program modules 660 may also transform the physical structure of the storage media.
• the specific transformation of physical structure may depend on various factors, in different implementations of this description. Examples of such factors may include, but are not limited to: the technology used to implement the storage media, whether the storage media are characterized as primary or secondary storage, or the like.
• the program modules 660 may transform the physical state of the semiconductor memory 620 when the software is encoded therein.
• the software may transform the state of transistors, capacitors, or other discrete circuit elements constituting the semiconductor memory 620.
• the storage media may be implemented using magnetic or optical technology such as drives 630.
• the program modules 660 may transform the physical state of magnetic or optical media, when the software is encoded therein. These transformations may include altering the magnetic characteristics of particular locations within given magnetic media. These transformations may also include altering the physical features or characteristics of particular locations within given optical media, to change the optical characteristics of those locations. It should be appreciated that various other transformations of physical media are possible without departing from the scope and spirit of the present description.
• the phrase "computer storage medium,” and variations thereof, does not include waves, signals, and/or other transitory and/or intangible communication media, per se.
• FIGURE 7 is a schematic diagram illustrating computer program products
• the example computer program product 700 for linear programming based decoding for memory devices arranged according to at least some embodiments presented herein.
• An illustrative embodiment of the example computer program product 700 is provided using a signal bearing medium 702, and may include at least one instruction 704.
• the at least one instruction 704 may include: one or more instructions for detecting a cell threshold voltage level of a memory cell of a flash memory device; one or more instructions for determining an interference voltage level of an interference cell that interferes with the memory cell; or one or more instructions for decoding the cell threshold voltage level in accordance with a set of beliefs to determine a value of the memory cell, the set of beliefs comprising a minimization of an objective function of a linear program representing inter-cell interference between the memory cell and the interference cell.
• the signal bearing medium 702 of the one or more computer program products 700 include a computer readable medium 706, a recordable medium 708, and/or a communications medium 710.
• program modules include routines, programs, components, data structures, and other types of structures that perform particular tasks or implement particular abstract data types.
• program modules include routines, programs, components, data structures, and other types of structures that perform particular tasks or implement particular abstract data types.
• program modules include routines, programs, components, data structures, and other types of structures that perform particular tasks or implement particular abstract data types.
• the subject matter described herein may be practiced with other computer system configurations, including hand-held devices, multi-core processor systems, microprocessor-based or programmable consumer electronics, minicomputers, mainframe computers, or the like.
• a range includes each individual member.
• a group having 1-3 elements refers to groups having 1 , 2, or 3 elements.
• a group having 1 -5 elements refers to groups having 1 , 2, 3, 4, or 5 elements, and so forth.

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