3 research outputs found
Fixed-point realization of fast nonlinear Fourier transform algorithm for FPGA implementation of optical data processing
The nonlinear Fourier transform (NFT) based signal processing has attracted considerable attention as a promising tool for fibre nonlinearity mitigation in optical transmission. However, the mathematical complexity of NFT algorithms and the noticeable distinction of the latter from the βconventionalβ (Fourier-based) methods make it difficult to adapt this approach for practical applications. In our work, we demonstrate a hardware implementation of the fast direct NFT operation: it is used to map the optical signal onto its nonlinear Fourier spectrum, i.e. to demodulate the data. The main component of the algorithm is the matrix-multiplier unit, implemented on field-programmable gate arrays (FPGA) and used in our study for the estimation of required hardware resources. To design the best performing implementation in limited resources, we carry out the processing accuracy analysis to estimate the optimal bit width. The fast NFT algorithm that we analyse, is based on the FFT, which leads to the O(N log^{2}_{2} N) methodβs complexity for the signal consisting of N samples. Our analysis revealed the significant demand in DSP blocks on the used board, which is caused by the complex-valued matrix operations and FFTs. Nevertheless, it seems to be possible to utilise further the parallelisation of our NFT-processing implementation for the more efficient NFT hardware realisation
Conveyorized Implementation of ASWM Image Filter on PLD
The object of research is the adaptive switching weighted median image filter (ASWM) algorithm. This algorithm is one of the most effective in the field of impulse noise suppression. The computational complexity and algorithmic features of this adaptive nonlinear filter make it impossible to implement a filter that works in real time on modern PLD microcircuits.
The most problematic areas of the algorithm are the weight coefficient estimation cycle, which has no limit on the number of iterations and contains a large number of division operations. This does not allow implementing the filter on PLDs with a sufficiently effective method.
In the course of the research, the programming model of the filter in Python was used. The performance of the algorithm was assessed using the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) metrics.
Modeling made it possible to find out empirically the number of iterations of the cycle for estimating the weight coefficients at different levels of noise density and to estimate the effect of artificial limitation of the maximum number of iterations on the filter performance. Regardless of the intensity of the noise impact, the algorithm performs less than 40 iterations of the evaluation cycle. Let's also simulate the operation of the algorithm with different variants of the division module implementation. The paper considers the main of them and offers the most optimal in terms of the ratio of accuracy/hardware costs for implementation. Thus, a modified algorithm was proposed that does not have these disadvantages.
Thanks to modifications of the algorithm, it is possible to implement a pipelined ASWM image filter on modern PLDs. The filter is synthesized for the main families of Intel PLDs. The implementation, which is not inferior in terms of SSIM and PSNR metrics to the original algorithm, requires less than 65,000 FPGA logical cells and allows filtering of monochrome images with FullHD resolution at 48 frames/s at a clock frequency of 100 MHz
ΠΠΎΠ΄ΠΈΡΡΠΊΠΎΠ²Π°Π½ΠΈΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΏΠΎΡΡΠΊΡ ΠΊΠΎΡΠ΅Π½ΡΠ² ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΡΠ² ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ ΠΏΡΠΈ Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ ΠΠ§Π₯ ΠΊΠΎΠ΄ΡΠ²
Modified algorithm for searching the roots of the error locators polynominal while decoding BCH codes / Krylova V. Π., Π’verytnykova, Π. Π., Vasylchenkov O. G., Kolisnyk T. P. // Radio Electronics, Computer Science, Control. β 2020. β β 3. β P. 150β157. β DOI: ttps://doi.org/10.15588/1607-3274-2020-3-14.Krylova V. Π., Π’verytnykova Π. Π., Vasylchenkov, O. G., & Kolisnyk, T. P. (2020). MODIFIED ALGORITHM FOR SEARCHING THE ROOTS OF THE ERROR LOCATORS POLYNOMINAL WHILE DECODING BCH CODES. Radio Electronics, Computer Science, Control, (3), 150β157. https://doi.org/10.15588/1607-3274-2020-3-14.ΠΠΊΡΡΠ°Π»ΡΠ½ΡΡΡΡ. Π£ ΡΠ΅Π»Π΅ΠΊΠΎΠΌΡΠ½ΡΠΊΠ°ΡΡΠΉΠ½ΠΈΡ
ΡΠ° ΡΠ½ΡΠΎΡΠΌΠ°ΡΡΠΉΠ½ΠΈΡ
ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
Π·Π²βΡΠ·ΠΊΡ Π· ΠΏΡΠ΄Π²ΠΈΡΠ΅Π½ΠΎΡ ΡΡΠΌΠΎΠ²ΠΎΡ ΡΠΊΠ»Π°Π΄ΠΎΠ²ΠΎΡ Π²ΠΈΠΊΠΎΡΠΈΡΡΠΎΠ²ΡΡΡΡΡΡ ΠΏΠ΅ΡΠ΅ΡΠΊΠΎΠ΄ΠΎΡΡΡΠΉΠΊΡ ΡΠΈΠΊΠ»ΡΡΠ½Ρ ΠΠ§Π₯ ΡΠ° ΠΊΠΎΠ΄ΠΈ Π ΡΠ΄Π°-Π‘ΠΎΠ»ΠΎΠΌΠΎΠ½Π°. ΠΠΎΡΠΈΠ³ΡΠ²Π°Π½Π½Ρ ΡΠ° Π²ΠΈΠΏΡΠ°Π²Π»Π΅Π½Π½Ρ ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ Π² ΠΏΠΎΠ²ΡΠ΄ΠΎΠΌΠ»Π΅Π½Π½Ρ
Π²ΠΈΠΌΠ°Π³Π°Ρ Π΅ΡΠ΅ΠΊΡΠΈΠ²Π½ΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄ΡΠ² Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ. ΠΠ΄Π½ΠΈΠΌ Π· Π΅ΡΠ°ΠΏΡΠ² ΠΏΡΠΎΡΠ΅Π΄ΡΡΠΈ Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ Π Π‘ Ρ ΠΠ§Π₯ ΠΊΠΎΠ΄ΡΠ² Π΄Π»Ρ Π²ΠΈΠ·Π½Π°ΡΠ΅Π½Π½Ρ ΠΏΠΎΠ·ΠΈΡΡΠΉ
ΡΠΏΠΎΡΠ²ΠΎΡΠ΅Π½Ρ Ρ ΠΏΠΎΡΡΠΊ ΠΊΠΎΡΠ΅Π½ΡΠ² ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΡΠ² ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ. ΠΠ±ΡΠΈΡΠ»Π΅Π½Π½Ρ ΠΊΠΎΡΠ΅Π½ΡΠ² ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π°, ΠΎΡΠΎΠ±Π»ΠΈΠ²ΠΎ Ρ ΠΊΠΎΠ΄ΡΠ² Π·Ρ Π·Π½Π°ΡΠ½ΠΎΡ
ΠΊΠΎΡΠ΅ΠΊΡΡΡ Π·Π΄Π°ΡΠ½ΡΡΡΡ, Ρ ΡΡΡΠ΄ΠΎΠΌΡΡΡΠΊΠΎΡ Π·Π°Π²Π΄Π°Π½Π½ΡΠΌ, ΡΠΎ Π²ΠΈΠΌΠ°Π³Π°Ρ Π²ΠΈΡΠΎΠΊΠΎΡ ΠΎΠ±ΡΠΈΡΠ»ΡΠ²Π°Π»ΡΠ½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΠΎΡΡΡ. Π’ΠΎΠΌΡ ΡΠ΄ΠΎΡΠΊΠΎΠ½Π°Π»Π΅Π½Π½Ρ
ΠΌΠ΅ΡΠΎΠ΄ΡΠ² Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ ΠΠ§Π₯ Ρ Π Π‘ ΠΊΠΎΠ΄ΡΠ², ΡΠΎ Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡΡΡ Π·ΠΌΠ΅Π½ΡΠΈΡΠΈ ΡΠΊΠ»Π°Π΄Π½ΡΡΡΡ ΠΎΠ±ΡΠΈΡΠ»Π΅Π½Ρ, Ρ Π°ΠΊΡΡΠ°Π»ΡΠ½ΠΈΠΌ Π·Π°Π²Π΄Π°Π½Π½ΡΠΌ.
ΠΠ΅ΡΠ° ΡΠΎΠ±ΠΎΡΠΈ. ΠΠΎΡΠ»ΡΠ΄ΠΆΠ΅Π½Π½Ρ Ρ ΡΠΈΠ½ΡΠ΅Π· ΠΏΡΠΈΡΠΊΠΎΡΠ΅Π½ΠΎΠ³ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΠΏΠΎΡΡΠΊΡ ΠΊΠΎΡΠ΅Π½ΡΠ² ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΡΠ² ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ, ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎΠ³ΠΎ Ρ Π²ΠΈΠ³Π»ΡΠ΄Ρ Π°ΡΡΠ½Π½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π° Π· ΠΊΠΎΠ΅ΡΡΡΡΡΠ½ΡΠ°ΠΌΠΈ Π² ΠΊΡΠ½ΡΠ΅Π²ΠΈΡ
ΠΏΠΎΠ»ΡΡ
, ΡΠΊΠΈΠΉ Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡ ΠΏΡΠΈΡΠΊΠΎΡΠΈΡΠΈ ΠΏΡΠΎΡΠ΅Ρ Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ ΠΠ§Π₯ Ρ
Π Π‘ ΠΊΠΎΠ΄ΡΠ².
ΠΠ΅ΡΠΎΠ΄. ΠΠ»Π°ΡΠΈΡΠ½ΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΡΡΠΊΡ ΠΊΠΎΡΠ΅Π½ΡΠ² Π½Π° Π±Π°Π·Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ Π§Π΅Π½Ρ Π²ΠΈΠΊΠΎΠ½ΡΡΡΡΡΡ Π·Π° Π΄ΠΎΠΏΠΎΠΌΠΎΠ³ΠΎΡ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠΊΠΈ ΠΊΡΠ½ΡΠ΅Π²ΠΈΡ
ΠΏΠΎΠ»ΡΠ²
ΠΠ°Π»ΡΠ° Ρ ΡΡΡΠ΄ΠΎΠΌΡΡΡΠΊΡΡΡΡ ΡΠΎΠ·ΡΠ°Ρ
ΡΠ½ΠΊΡΠ², Π² Π΄Π°Π½ΠΎΠΌΡ Π²ΠΈΠΏΠ°Π΄ΠΊΡ, Π·Π°Π»Π΅ΠΆΠΈΡΡ Π²ΡΠ΄ ΠΊΡΠ»ΡΠΊΠΎΡΡΡ ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ Π΄ΠΎΠ΄Π°Π²Π°Π½Π½Ρ Ρ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ. ΠΠ»Ρ Π»ΠΈΠ½Π΅Π°ΡΠΈΠ·ΠΈΠ²Π°Π½ΠΈΡ
ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΡΠ² ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ° ΠΏΠΎΡΡΠΊΡ ΠΊΠΎΡΠ΅Π½ΡΠ², Π·Π°ΡΠ½ΠΎΠ²Π°Π½Π° Π½Π° Π΄Π²ΡΠΉΠΊΠΎΠ²ΡΠΉ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΡΡ ΡΠ° Π·Π΄ΡΠΉΡΠ½ΡΡΡΡΡΡ Π· ΡΡΠ°Ρ
ΡΠ²Π°Π½Π½ΡΠΌ Π·Π½Π°ΡΠ΅Π½Ρ
ΠΎΡΡΠΈΠΌΠ°Π½ΠΈΡ
Π½Π° ΠΏΠΎΠΏΠ΅ΡΠ΅Π΄Π½ΡΡ
Π΅ΡΠ°ΠΏΠ°Ρ
ΠΎΠ±ΡΠΈΡΠ»Π΅Π½Π½Ρ, ΡΠΎ Π·Π°Π±Π΅Π·ΠΏΠ΅ΡΡΡ ΠΌΡΠ½ΡΠΌΠ°Π»ΡΠ½Π΅ ΡΠΈΡΠ»ΠΎ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΡΠ½ΠΈΡ
ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ.
Π Π΅Π·ΡΠ»ΡΡΠ°ΡΠΈ. Π ΠΎΠ·ΡΠΎΠ±Π»Π΅Π½ΠΎ ΠΏΡΠΈΡΠΊΠΎΡΠ΅Π½ΠΈΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΎΠ±ΡΠΈΡΠ»Π΅Π½Π½Ρ Π·Π½Π°ΡΠ΅Π½Ρ ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΡΠ² ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ Ρ Π²ΡΡΡ
ΡΠΎΡΠΊΠ°Ρ
ΠΊΡΠ½ΡΠ΅Π²ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ GF (2m) Π΄Π»Ρ Π»ΠΈΠ½Π΅Π°ΡΠΈΠ·ΠΈΡΠΎΠ²Π°Π½ΠΈΡ
ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½ΡΠ² Π½Π° Π±Π°Π·Ρ ΠΌΠ΅ΡΠΎΠ΄Ρ ΠΠ΅ΡΠ»Π΅ΠΊΠ΅ΠΌΠΏΠ°-ΠΠ΅ΡΡΡ. ΠΠ»Π³ΠΎΡΠΈΡΠΌ ΠΌΡΡΡΠΈΡΡ ΠΌΡΠ½ΡΠΌΠ°Π»ΡΠ½Ρ
ΠΊΡΠ»ΡΠΊΡΡΡΡ ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ Π΄ΠΎΠ΄Π°Π²Π°Π½Ρ, Π·Π° ΡΠ°Ρ
ΡΠ½ΠΎΠΊ Π²ΠΈΠΊΠΎΡΠΈΡΡΠ°Π½Π½Ρ Π½Π° ΠΊΠΎΠΆΠ½ΠΎΠΌΡ Π΅ΡΠ°ΠΏΡ ΠΎΠ±ΡΠΈΡΠ»Π΅Π½Ρ, Π·Π½Π°ΡΠ΅Π½Ρ ΠΎΡΡΠΈΠΌΠ°Π½ΠΈΡ
Π½Π° ΠΏΠΎΠΏΠ΅ΡΠ΅Π΄Π½ΡΠΎΠΌΡ
ΠΊΡΠΎΡΡ, Π° ΡΠ°ΠΊΠΎΠΆ Π²ΠΈΠΊΠΎΠ½Π°Π½Π½Ρ ΡΠΊΠ»Π°Π΄Π°Π½Π½Ρ Π² ΠΊΡΠ½ΡΠ΅Π²ΠΎΠΌΡ ΠΏΠΎΠ»Ρ GF(2). ΠΠ°ΠΏΡΠΎΠΏΠΎΠ½ΠΎΠ²Π°Π½ΠΎ ΠΌΠΎΠ΄ΠΈΡΡΠΊΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΡΡΠΊΡ ΠΊΠΎΡΠ΅Π½ΡΠ² Π΄Π»Ρ
Π°ΡΡΠ½Π½ΠΈΡ
ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΡΠ² Π½Π°Π΄ ΠΊΡΠ½ΡΠ΅Π²ΠΈΠΌΠΈ ΠΏΠΎΠ»ΡΠΌΠΈ, ΡΠΎ Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡ Π²ΠΈΠ·Π½Π°ΡΠΈΡΠΈ ΠΏΠΎΠ·ΠΈΡΡΡ ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ Π² ΠΊΠΎΠ΄ΠΎΠ²ΠΎΠΌΡ ΡΠ»ΠΎΠ²Ρ ΠΏΡΠ΄ ΡΠ°Ρ Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ
ΡΠΈΠΊΠ»ΡΡΠ½ΠΈΡ
ΠΠ§Π₯ Ρ Π Π‘ ΠΊΠΎΠ΄ΡΠ².
ΠΠΈΡΠ½ΠΎΠ²ΠΊΠΈ. ΠΠ°ΡΠΊΠΎΠ²Π° Π½ΠΎΠ²ΠΈΠ·Π½Π° ΡΠΎΠ±ΠΎΡΠΈ ΠΏΠΎΠ»ΡΠ³Π°Ρ Π² ΡΠ΄ΠΎΡΠΊΠΎΠ½Π°Π»Π΅Π½Π½Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΠΎΠ±ΡΠΈΡΠ»Π΅Π½Π½Ρ ΠΊΠΎΡΠ΅Π½ΡΠ² ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π° Π»ΠΎΠΊΠ°ΡΠΎΡΡΠ² ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ, ΠΊΠΎΠ΅ΡΡΡΡΡΠ½ΡΠΈ ΡΠΊΠΎΠ³ΠΎ Π½Π°Π»Π΅ΠΆΠ°ΡΡ Π΄ΠΎ Π΅Π»Π΅ΠΌΠ΅Π½ΡΡΠ² ΠΊΡΠ½ΡΠ΅Π²ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ. ΠΡΠΈ ΡΡΠΎΠΌΡ ΡΠΏΡΠΎΡΡΡΡΡΡΡ ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ° Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ ΡΠΈΠΊΠ»ΡΡΠ½ΠΈΡ
ΠΠ§Π₯ Ρ Π Π‘ ΠΊΠΎΠ΄ΡΠ², Π·Π° ΡΠ°Ρ
ΡΠ½ΠΎΠΊ Π·Π½ΠΈΠΆΠ΅Π½Π½Ρ ΠΎΠ±ΡΠΈΡΠ»ΡΠ²Π°Π»ΡΠ½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΠΎΡΡΡ ΠΎΠ΄Π½ΠΎΠ³ΠΎ Π· Π΅ΡΠ°ΠΏΡΠ² Π΄Π΅ΠΊΠΎΠ΄ΡΠ²Π°Π½Π½Ρ β Π·Π½Π°Ρ
ΠΎΠ΄ΠΆΠ΅Π½Π½Ρ ΠΏΠΎΠ·ΠΈΡΡΠΉ ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ Π· Π²ΠΈΠΊΠΎΡΠΈΡΡΠ°Π½Π½ΡΠΌ ΠΌΠΎΠ΄ΠΈΡΡΠΊΠΎΠ²Π°Π½ΠΎΠ³ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΠΠ΅ΡΠ»Π΅ΠΊΠ΅ΠΌΠΏΠ°-ΠΠ΅ΡΡΡ. ΠΠ°Π½Ρ ΡΠ°ΠΊΡΠΈ ΠΏΡΠ΄ΡΠ²Π΅ΡΠ΄ΠΆΠ΅Π½Ρ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ°ΠΌΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠ½ΠΎΠ³ΠΎ
ΠΌΠΎΠ΄Π΅Π»ΡΠ²Π°Π½Π½Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΠΏΠΎΡΡΠΊΡ ΠΊΠΎΡΠ΅Π½ΡΠ² ΠΏΠΎΠ»ΡΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΡΠ² ΠΏΠΎΠΌΠΈΠ»ΠΎΠΊ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΠΎ Π·Π°ΡΡΠΎΡΡΠ²Π°Π½Π½Ρ ΠΏΡΠΈΡΠΊΠΎΡΠ΅Π½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Ρ
Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡ Π΄ΠΎΡΡΠ³ΡΠΈ Π²ΠΈΠ³ΡΠ°ΡΡ ΠΏΠΎ ΡΠ²ΠΈΠ΄ΠΊΠΎΠ΄ΡΡ Π² 1,5 ΡΠ°Π·ΠΈ.Context. In telecommunications and information systems with an increased noise component the noise-resistant cyclic BCH and
Reed-Solomon codes are used. The adjustment and correcting errors in a message require some effective decoding methods. One of
the stages in the procedure of decoding RS and BCH codes to determine the position of distortions is the search for the roots of the
error locator polynomial. The calculation of polynomial roots, especially for codes with significant correction capacity is a laborious
task requiring high computational complexity. That is why the improvement of BCH and RS codes decoding methods providing to
reduce the computational complexity is an urgent task.
Objective. The investigation and synthesis of the accelerated roots search algorithm of the error locator polynomial presented as
an affine polynomial with coefficients in the finite fields, which allows accelerating the process of BCH and RS code decoding.
Method. The classical roots search method based on the Chanβs algorithm is performed using the arithmetic of the Galois finite
fields and the laborious calculation, in this case depends on the number of addition and multiplication operations. For linearized polynomials, the roots search procedure based on binary arithmetic is performed taking into account the values obtained at the previous
stages of the calculation, which provides the minimum number of arithmetic operations.
Results. An accelerated algorithm for calculating the values of the error locator polynomial at all points of the GF(2m) finite field
for linearized polynomials based on the Berlekamp-Massey method has been developed. The algorithm contains a minimum number
of addition operations, due to the use at each stage of the calculations the values obtained at the previous step, as well as the addition
in the finite field GF(2). A modified roots search method for affine polynomials over the finite fields has been proposed to determine
error positions in the code word while decoding the cyclic BCH and RS codes.
Conclusions. The scientific newness of the work is to improve the algorithm of calculating the roots of the error locator polynomial, which coefficients belong to the elements of the finite field. At the same time it simplifies the procedure for cyclic BCH and RS
codes decoding, due to reducing the computational complexity of one of the decoding stages, especially finding the error positions
using the modified Berlekamp-Massey algorithm. These facts are confirmed by the simulation program results of the roots search of
the error locator polynomial algorithm. It is shown, that the application of the accelerated method permits to reach a gain on speed of
1.5 times.ΠΠΊΡΡΠ°Π»ΡΠ½ΠΎΡΡΡ. Π ΡΠ΅Π»Π΅ΠΊΠΎΠΌΠΌΡΠ½ΠΈΠΊΠ°ΡΠΈΠΎΠ½Π½ΡΡ
ΠΈ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΎΠ½Π½ΡΡ
ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
ΡΠ²ΡΠ·ΠΈ Ρ ΠΏΠΎΠ²ΡΡΠ΅Π½Π½ΠΎΠΉ ΡΡΠΌΠΎΠ²ΠΎΠΉ ΡΠΎΡΡΠ°Π²Π»ΡΡΡΠ΅ΠΉ
ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΠΏΠΎΠΌΠ΅Ρ
ΠΎΡΡΡΠΎΠΉΡΠΈΠ²ΡΠ΅ ΡΠΈΠΊΠ»ΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΠ§Π₯ ΠΈ ΠΊΠΎΠ΄Ρ Π ΠΈΠ΄Π°-Π‘ΠΎΠ»ΠΎΠΌΠΎΠ½Π°. ΠΠΎΡΡΠ΅ΠΊΡΠΈΡΠΎΠ²ΠΊΠ° ΠΈ ΠΈΡΠΏΡΠ°Π²Π»Π΅Π½ΠΈΠ΅ ΠΎΡΠΈΠ±ΠΎΠΊ Π² ΡΠΎΠΎΠ±ΡΠ΅Π½ΠΈΠΈ ΡΡΠ΅Π±ΡΠ΅Ρ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ. ΠΠ΄Π½ΠΈΠΌ ΠΈΠ· ΡΡΠ°ΠΏΠΎΠ² ΠΏΡΠΎΡΠ΅Π΄ΡΡΡ Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ Π Π‘ ΠΈ ΠΠ§Π₯ ΠΊΠΎΠ΄ΠΎΠ² Π΄Π»Ρ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ ΠΏΠΎΠ·ΠΈΡΠΈΠΉ ΠΈΡΠΊΠ°ΠΆΠ΅Π½ΠΈΠΉ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΏΠΎΠΈΡΠΊ ΠΊΠΎΡΠ½Π΅ΠΉ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΠΎΠ² ΠΎΡΠΈΠ±ΠΎΠΊ. ΠΡΡΠΈΡΠ»Π΅Π½ΠΈΠ΅ ΠΊΠΎΡΠ½Π΅ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π°, ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎ Ρ ΠΊΠΎΠ΄ΠΎΠ² ΡΠΎ Π·Π½Π°ΡΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΠΊΠΎΡΡΠ΅ΠΊΡΠΈΡΡΡΡΠ΅ΠΉ ΡΠΏΠΎΡΠΎΠ±Π½ΠΎΡΡΡΡ, ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΡΡΠ΄ΠΎΠ΅ΠΌΠΊΠΎΠΉ Π·Π°Π΄Π°ΡΠ΅ΠΉ, ΡΡΠ΅Π±ΡΡΡΠ΅ΠΉ Π²ΡΡΠΎΠΊΠΎΠΉ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΠΈ. ΠΠΎΡΡΠΎΠΌΡ ΡΡΠΎΠ²Π΅ΡΡΠ΅Π½ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΠ§Π₯ ΠΈ Π Π‘ ΠΊΠΎΠ΄ΠΎΠ², ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΡ
ΡΠΌΠ΅Π½ΡΡΠΈΡΡ
ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΡ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ, ΡΠ²Π»ΡΠ΅ΡΡΡ Π°ΠΊΡΡΠ°Π»ΡΠ½ΠΎΠΉ Π·Π°Π΄Π°ΡΠ΅ΠΉ.
Π¦Π΅Π»Ρ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ. ΠΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΈ ΡΠΈΠ½ΡΠ΅Π· ΡΡΠΊΠΎΡΠ΅Π½Π½ΠΎΠ³ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° ΠΏΠΎΠΈΡΠΊΠ° ΠΊΠΎΡΠ½Π΅ΠΉ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΠΎΠ² ΠΎΡΠΈΠ±ΠΎΠΊ, ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Π½ΠΎΠ³ΠΎ Π² Π²ΠΈΠ΄Π΅ Π°ΡΡΠΈΠ½Π½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π° Ρ ΠΊΠΎΡΡΡΠΈΡΠΈΠ΅Π½ΡΠ°ΠΌΠΈ Π² ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
ΠΏΠΎΠ»ΡΡ
, ΠΊΠΎΡΠΎΡΡΠΉ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΡΠΊΠΎΡΠΈΡΡ ΠΏΡΠΎΡΠ΅ΡΡ Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΠ§Π₯ ΠΈ Π Π‘ ΠΊΠΎΠ΄ΠΎΠ².
ΠΠ΅ΡΠΎΠ΄. ΠΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΠΈΡΠΊΠ° ΠΊΠΎΡΠ½Π΅ΠΉ Π½Π° Π±Π°Π·Π΅ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° Π§Π΅Π½Ρ Π²ΡΠΏΠΎΠ»Π½ΡΠ΅ΡΡΡ Ρ ΠΏΠΎΠΌΠΎΡΡΡ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠΊΠΈ ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
ΠΏΠΎΠ»Π΅ΠΉ ΠΠ°Π»ΡΠ° ΠΈ ΡΡΡΠ΄ΠΎΠ΅ΠΌΠΊΠΎΡΡΡ ΡΠ°ΡΡΠ΅ΡΠΎΠ², Π² Π΄Π°Π½Π½ΠΎΠΌ ΡΠ»ΡΡΠ°Π΅, Π·Π°Π²ΠΈΡΠΈΡ ΠΎΡ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ ΡΠ»ΠΎΠΆΠ΅Π½ΠΈΡ ΠΈ ΡΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΡ. ΠΠ»Ρ Π»ΠΈΠ½Π΅Π°ΡΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΎΠ² ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ° ΠΏΠΎΠΈΡΠΊΠ° ΠΊΠΎΡΠ½Π΅ΠΉ, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½Π°Ρ Π½Π° Π΄Π²ΠΎΠΈΡΠ½ΠΎΠΉ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠΊΠ΅, ΠΎΡΡΡΠ΅ΡΡΠ²Π»ΡΠ΅ΡΡΡ Ρ ΡΡΠ΅ΡΠΎΠΌ Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ
ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΡ
Π½Π° ΠΏΡΠ΅Π΄ΡΠ΄ΡΡΠΈΡ
ΡΡΠ°ΠΏΠ°Ρ
Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ, ΡΡΠΎ ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠΈΠ²Π°Π΅Ρ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ.
Π Π΅Π·ΡΠ»ΡΡΠ°ΡΡ. Π Π°Π·ΡΠ°Π±ΠΎΡΠ°Π½ ΡΡΠΊΠΎΡΠ΅Π½Π½ΡΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΠΎΠ² ΠΎΡΠΈΠ±ΠΎΠΊ Π²ΠΎ Π²ΡΠ΅Ρ
ΡΠΎΡΠΊΠ°Ρ
ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ GF(2m) Π΄Π»Ρ Π»ΠΈΠ½Π΅Π°ΡΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½ΠΎΠ² Π½Π° Π±Π°Π·Π΅ ΠΌΠ΅ΡΠΎΠ΄Π° ΠΠ΅ΡΠ»Π΅ΠΊΡΠΌΠΏΠ°-ΠΠ΅ΡΡΠΈ. ΠΠ»Π³ΠΎΡΠΈΡΠΌ ΡΠΎΠ΄Π΅ΡΠΆΠΈΡ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ ΡΠ»ΠΎΠΆΠ΅Π½ΠΈΠΉ, Π·Π° ΡΡΠ΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ Π½Π° ΠΊΠ°ΠΆΠ΄ΠΎΠΌ ΡΡΠ°ΠΏΠ΅ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ, Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΡ
Π½Π° ΠΏΡΠ΅Π΄ΡΠ΄ΡΡΠ΅ΠΌ ΡΠ°Π³Π΅, Π° ΡΠ°ΠΊΠΆΠ΅ Π²ΡΠΏΠΎΠ»Π½Π΅Π½ΠΈΡ ΡΠ»ΠΎΠΆΠ΅Π½ΠΈΡ Π² ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΌ ΠΏΠΎΠ»Π΅ GF(2). ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ ΠΌΠΎΠ΄ΠΈΡΠΈΡΠΈΡΠΎΠ²Π°Π½Π½ΡΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΠΈΡΠΊΠ° ΠΊΠΎΡΠ½Π΅ΠΉ Π΄Π»Ρ
Π°ΡΡΠΈΠ½Π½ΡΡ
ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΎΠ² Π½Π°Π΄ ΠΊΠΎΠ½Π΅ΡΠ½ΡΠΌΠΈ ΠΏΠΎΠ»ΡΠΌΠΈ, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠΈΠΉ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ ΠΏΠΎΠ·ΠΈΡΠΈΠΈ ΠΎΡΠΈΠ±ΠΎΠΊ Π² ΠΊΠΎΠ΄ΠΎΠ²ΠΎΠΌ ΡΠ»ΠΎΠ²Π΅ ΠΏΡΠΈ Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΡΠΈΠΊΠ»ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΠ§Π₯ ΠΈ Π Π‘ ΠΊΠΎΠ΄ΠΎΠ².
ΠΡΠ²ΠΎΠ΄Ρ. ΠΠ°ΡΡΠ½Π°Ρ Π½ΠΎΠ²ΠΈΠ·Π½Π° ΡΠ°Π±ΠΎΡΡ ΡΠΎΡΡΠΎΠΈΡ Π² ΡΡΠΎΠ²Π΅ΡΡΠ΅Π½ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΠΈ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ ΠΊΠΎΡΠ½Π΅ΠΉ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π° Π»ΠΎΠΊΠ°ΡΠΎΡΠΎΠ²
ΠΎΡΠΈΠ±ΠΎΠΊ, ΠΊΠΎΡΡΡΠΈΡΠΈΠ΅Π½ΡΡ ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ ΠΏΡΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°Ρ ΡΠ»Π΅ΠΌΠ΅Π½ΡΠ°ΠΌ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ. ΠΡΠΈ ΡΡΠΎΠΌ ΡΠΏΡΠΎΡΠ°Π΅ΡΡΡ ΠΏΡΠΎΡΠ΅Π΄ΡΡΠ° Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ
ΡΠΈΠΊΠ»ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΠ§Π₯ ΠΈ Π Π‘ ΠΊΠΎΠ΄ΠΎΠ², Π·Π° ΡΡΠ΅Ρ ΡΠ½ΠΈΠΆΠ΅Π½ΠΈΡ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΠΈ ΠΎΠ΄Π½ΠΎΠ³ΠΎ ΠΈΠ· ΡΡΠ°ΠΏΠΎΠ² Π΄Π΅ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ β Π½Π°Ρ
ΠΎΠΆΠ΄Π΅Π½ΠΈΡ
ΠΏΠΎΠ·ΠΈΡΠΈΠΉ ΠΎΡΠΈΠ±ΠΎΠΊ Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ ΠΌΠΎΠ΄ΠΈΡΠΈΡΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠ³ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° ΠΠ΅ΡΠ»Π΅ΠΊΡΠΌΠΏΠ°-ΠΠ΅ΡΡΠΈ. ΠΠ°Π½Π½ΡΠ΅ ΡΠ°ΠΊΡΡ ΠΏΠΎΠ΄ΡΠ²Π΅ΡΠΆΠ΄Π΅Π½Ρ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ°ΠΌΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠ½ΠΎΠ³ΠΎ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° ΠΏΠΎΠΈΡΠΊΠ° ΠΊΠΎΡΠ½Π΅ΠΉ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠ° Π»ΠΎΠΊΠ°ΡΠΎΡΠΎΠ² ΠΎΡΠΈΠ±ΠΎΠΊ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅
ΡΡΠΊΠΎΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Π° ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ Π΄ΠΎΡΡΠΈΡΡ Π²ΡΠΈΠ³ΡΡΡΠ° ΠΏΠΎ Π±ΡΡΡΡΠΎΠ΄Π΅ΠΉΡΡΠ²ΠΈΡ Π² 1,5 ΡΠ°Π·Π°