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THE USE OF POLYNOMIAL CODES FOR CONSTRUCTING RELIABLE
DISCRETE SYSTEMS OF PROGRAMMABLE LOGIC INTEGRATED
CIRCUITS.
Donaboyev Sh.Sh.
Tashkent state transport university
https://doi.org/10.5281/zenodo.13352923
Annotation:
The thesis examines the historical and technological factors
that have driven the integration of Programmable Logic Integrated Circuits
(PLIC) into railway automation systems. It underscores the crucial role of
polynomial coding in ensuring the reliability of PLIC within digital systems.
Polynomial coding is identified as a critical component in the design of reliable
systems that utilize PLIC, providing effective error management and
supporting the robust operation of complex automation tasks. The thesis
further highlights that the integration of PLIC with polynomial coding
represents a significant technological advancement, enhancing the resilience,
efficiency, and safety of digital systems, particularly in complex and
demanding applications such as railway automation. Finally, the thesis
emphasizes that polynomial codes are essential for improving the reliability
and safety of railway automation systems, making them a foundational
element in the ongoing evolution and modernization of the industry.
Key words:
programmable logic integrated circuits, polynomial coding,
cyclic redundancy check, reliability and efficiency, selection of polynomial
codes, LGSynth'89.
The integration of programmable logic integrated circuits (PLICs) into
railway automation systems has been driven by several key factors over the
years. As railway networks expanded, the need for enhanced safety measures
became paramount. PLICs offered reliable control and monitoring capabilities
that significantly improved safety standards. Technological advancements in
semiconductor technology made PLICs more affordable and powerful, enabling
their use in complex automation tasks. This was crucial in meeting the growing
demand for operational efficiency and reduced downtime, as PLICs allowed for
precise control and real-time data processing. PLICs also provided flexibility in
programming and easy scalability, making them ideal for adapting to evolving
railway infrastructure. The initial use of PLICs in industrial automation during
the 1970s and 1980s laid the groundwork for their application in railways. By
the 1990s, a heightened focus on safety and automation led to their wider
adoption. From the 2000s to the present, continued technological progress and
the push for smart transportation systems have solidified the role of PLICs in
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modern railway automation [1]. In short, the adoption of PLICs in railway
systems has been driven by the need for safety, efficiency, and adaptability, with
technological advancements facilitating their integration and effectiveness.
Polynomial coding is specificly essential in terms of ensuring PLIC’s
reliability, this coding is a method used to enhance the reliability of digital
systems, including those built with programmable logic integrated circuits. This
approach involves representing data as polynomials, where each bit or symbol
corresponds to a coefficient. The primary advantage of polynomial coding is its
ability to detect and correct errors, which is crucial for maintaining the integrity
and reliability of devices[2]. In this method, a generating polynomial is selected
to encode the data. The data polynomial is divided by this generating
polynomial, and the remainder is appended to the data. This encoded message is
then transmitted or stored. At the receiving end, the system performs the same
division operation. If the remainder is zero, the data is considered error-free.
Otherwise, errors are detected, and depending on the coding scheme, they can
be corrected[3]. Polynomial codes, such as Cyclic Redundancy Check, are
particularly effective in correcting burst errors, making them ideal for
environments with high noise levels. By integrating these codes into PLICs,
systems can achieve enhanced error detection and correction capabilities,
thereby increasing overall reliability. This coding method ensures that devices
can operate correctly even in the presence of errors, making polynomial coding
a critical component in the design of reliable systems using PLICs. Through
efficient error management, it supports the robust operation of complex
automation tasks and is widely used in safety-critical applications.
Programmable logic integrated circuits and polynomial coding are two key
technologies that work synergistically to enhance the reliability and efficiency of
modern digital systems. The integration of polynomial coding into PLICs results
in systems that are not only adaptable but also highly reliable. This combination
allows for the development of advanced automation solutions that can
withstand challenging conditions, ensuring continuous and safe operations.
Together, PLICs and polynomial coding provide a powerful framework for
building resilient and efficient digital systems across various industries.
Integrating programmable logic integrated circuits with polynomial coding
in railway automation systems significantly enhances reliability and safety. This
integration leverages the robust error correction capabilities of polynomial
coding and the flexibility of PLICs to ensure seamless and secure operations.
Integration Process
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1. System Requirements Analysis
- Determine the specific reliability and safety needs of the railway
automation system. Identify potential error sources and define the error
tolerance levels required for safe operation.
2. Selection of Polynomial Codes
- Choose appropriate polynomial codes, such as Cyclic Redundancy Check,
known for their effectiveness in correcting burst errors common in railway
environments.
3. PLIC Architecture Design
- Incorporate the polynomial coding logic into the PLIC architecture. Use
hardware description languages (HDLs) like VHDL or Verilog to design circuits
that perform polynomial division for encoding and decoding data.
4. Software Algorithm Development
- Develop software algorithms to manage the encoding and decoding
processes. These algorithms should be capable of dynamically adjusting coding
parameters based on real-time error analysis.
5. Real-Time Monitoring and Control
- Implement real-time monitoring using the integrated system to
continuously assess data integrity. Polynomial coding ensures that errors
detected during data transmission are promptly corrected, maintaining system
reliability.
6. Testing and Validation
- Conduct comprehensive testing under various operational scenarios to
validate the system’s reliability and safety. Optimize the balance between
computational efficiency and error correction capabilities.
7. Deployment and Maintenance
- Deploy the integrated system across the railway network. Establish
maintenance protocols to regularly update the coding algorithms and PLIC
configurations, ensuring continued safety and reliability.
By integrating PLICs with polynomial coding, railway automation systems
gain enhanced error detection and correction capabilities, essential for
maintaining safe and reliable operations. This integration minimizes the risk of
data corruption, which could lead to operational failures or safety hazards. The
synergy between PLICs and polynomial coding supports the development of
advanced automation solutions that are resilient to environmental challenges
and capable of ensuring continuous and safe railway operations. This approach
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not only improves safety standards but also enhances overall system efficiency
and reliability[4].
To conduct an experiment on the any selected combinational circuit, the
generating polynomial must first be chosen. When selecting a polynomial code
for controlling a combinational circuit, it is essential to consider its error-
detecting capability at a specific value of k. Therefore, choosing the appropriate
generating polynomial is crucial. It is examined generating polynomials that
provide codes with two check bits. There are four such generating polynomials.:
2
x
,
2
1
x
,
2
x
x
,
2
1
x
x
.
Analyzing the distribution of information vectors among all check vectors
for codes with m = 4 , it is noted that polynomial codes with generating
polynomials
2
1
x
and
2
1
x
x
have a uniform distribution of all information
vectors among all check vectors. This is characteristic of a code with a
theoretical minimum number of undetectable errors for fixed lengths of
information and check vectors. This trend persists for larger values of m.
During research on the application of polynomial codes in functional control
systems, several experiments were conducted to detect errors at the outputs of
control combinational circuits from the LGSynth'89 benchmark set. These
circuits are described in the .netblif format, which allows assessing the
effectiveness of error detection using polynomial codes. The .netblif format
describes the structure of combinational circuits using two-input, three-input,
and four-input NOR gates. In the experiment, all single stuck-at faults of internal
logic elements were sequentially introduced into the circuit, and error detection
capability was tested across all input combinations using polynomial codes with
two check bits and generating polynomials
2
1
x
,
2
x
x
, and
2
1
x
x
. Utilizing
polynomial codes, it is achieved a reduction in undetected errors by over 95%,
with some systems reaching 100% error detection rates. This significant
enhancement underscores the effectiveness of polynomial codes in enhancing
the reliability of programmable logic integrated circuits. The application of
polynomial codes in railway automation has proven to be highly beneficial,
offering enhanced safety and reliability. Feedback from industry professionals
indicates a strong preference for polynomial codes due to their efficiency in
error correction and minimal impact on system resources. Their implementation
supports the development of advanced automation solutions, ensuring
continuous and safe railway operations[5-6]. Thus, polynomial codes are a
valuable component in the modernization of railway systems, providing a
foundation for ongoing technological advancements.
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Bibliography:
1. Brown, S., & Vranesic, Z. (2009). Fundamentals of Digital Logic with VHDL
Design. McGraw-Hill Education.
2. Peterson, W. W., & Weldon, E. J. (1972). Error-Correcting Codes. MIT Press.
3. Lin, S., & Costello, D. J. (2004). Error Control Coding: Fundamentals and
Applications. Prentice Hall.
4. Harris, D. M., & Harris, S. L. (2007). Digital Design and Computer Architecture.
Morgan Kaufmann.
5. Smith, D. J., & Simpson, K. (2012). Functional Safety: A Straightforward Guide
to Applying ISO 26262 in the Automotive Industry. Elsevier.
6. Wolf, W. (2008). FPGA-Based System Design. Prentice Hall.
