TW384425B - Method and apparatus for solving polynomial of key function when decoding error correction codes - Google Patents

Abstract

articularly, a minimum amount of FFMs (Finite Field Multiplier), rather than an effective arrangement of FFIs (Finite Field Inverter) would be needed. The use of this new methods would need only 3 FFM's, without effective saving of FFI's, for calculation of Berlekamp Massey. The method and apparatus of the present invention may be widely used on the RS and BCH codes of proper encoding length.

Description

A7 B7 Central Standard of the Ministry of Economic Affairs: Industrial and consumer cooperation Du Yinxuan 5. Description of the invention (1) The present invention relates to a method and a device for decoding edge correction error codes (Eγγ〇γ Correcting Codes), and in particular, it relates to decoding In the process of the error correction code, a method and a device for determining an error locator polynomial (Eγγ〇γ Locator Polynomials) and an evaluator polynomial (Eγγ〇γ Evaluator Polynomials) are determined. Through a variety of different media, from the transmission position to the data transmission wheel in the destination, due to the transmission path and the "noise caused by the media itself, it will cause the transmission of data. The transmission of the data will not be the same as the received data. Same data. In order to determine the error of receiving data, various methods and techniques have been developed to detect and correct the error of received data. One method is to generate a message part (data transmitted) and a parity part. Copy (Parity Part, according to which information is recorded and corrected) code word (here). In this article, 'code word is obtained by performing an encoding operation on the original data. The code word has the same form and includes N The information of the symbols, in which the first K symbols are information symbols, and the N_κ symbols after the rain are odd supply symbols. Among all well-known error correction codes, BCH-Chaudhuri-Hocquenghen Codes and RS-Reed -Solomon Codes) is the most widely used Block Codes in the communication field and storage system applications. The basis of the mathematical examination of block scorpions is "ER Berlekamp, Algebraic Coding The ory, McGraw-Hill, New York, 1968 "and" S. Lin and DJ Costello, Wurror Cowiro / Coding: Fundamentals and Applications ^ Prentice-Hall, this paper size applies to the Chinese National Standard (CNS) A4 specification (2 丨 0 '〆297mm） (please read the note on the back of the page and then fill in this page) ic.? Ί -c. Duyin A7 B7, Shellfish Consumer Cooperation of the Central Sample Bureau of Zhongji Department (2 )

Englewood Cliffs, NJ, 1983 "both have detailed explanations. An (N, K) BCH or RS code has K message symbols and N coding symbols, where each symbol belongs to a GF (q) set of a BCH code or a GF (qm) set of an RS code. In the case of N = 2m-bu and N-K < mt, a binary (N, K) BCH code can correct t error symbols. In the case of 2t + p < N-K, a binary (N, K) RS code can correct t error symbols and p erasure symbols. For binary BCH codes, an error symbol can be easily corrected by finding the location of the error symbol. For RS codes, an error symbol can be corrected by finding the location of the error symbol and its error value. In the application of RS code, erasure is defined as an error at a known error position. Therefore, correcting this error is simplified to find the wrong value. Using the general RS decoder architecture, the method steps applied to error correction can be summarized as the following four steps: (1) Calculate the symdrome from the received codeword, (2) Calculate the error locator polynomial and error The evaluator polynomial, (3) finds the location of the error, and (4) calculates the error value. Assuming that errors and erasures are corrected, the four steps are corrected as follows: (1) Calculate the symptoms and Forney symptoms from the received codeword and erasure position, (2) Calculate the errors and erase the locator polynomial and Error and erasure evaluator polynomials, (3) where the error was found, and (4) the correction value for the correction and erasure. Referring to circle la, it shows the general decoding steps. The received data R (x) is input into the Symptom Calculator 10 to generate the Symptom Polynomial S (x). This paper size applies the Chinese National Standard (CNS) A4 specification (210X297 mm) --------- C- ----- iT ------ c (please «* read the notes on the back side before filling out this page) A7 B7 V. Description of the invention (3) It represents the wrong type of code word to correct the error. Symptoms depend only on the type of error, not the codeword transmitted. Next, the symptoms are input into a Key Equation Solver 12 and the well-known Berlekamp-Massey algorithm is used to generate a recorded error locator polynomial σ (χ) and an error evaluator polynomial Ω (χ). The error locator polynomial indicates where the error occurred, and the error evaluator polynomial indicates the value of the error. In the next step, the error determinator polynomial is passed to the Chien searcher 14 to solve the root p; 1 of the equation, which represents the position of the error symbol. The error evaluator 16 receives the root p; 1 and the error polynomial Ω (χ) to generate an error value corresponding to the root acridine. When implementing the chain equation solver (the second step above), this step is related to solving the following healthy equations: S (x) σ (χ) = Ω (χ) mod xN "K; where S (x ) Is an indication polynomial, σ (χ) is an error locator polynomial, and Ω (χ) is an error evaluator polynomial. When errors and erasures are corrected simultaneously, σ (χ) and Ω (χ) are the error and erasure locator polynomials and the wrong supply and erasure evaluator polynomials, respectively, where σ (χ) = λ (χ ) Λ (χ), and λ (χ) and Λ (χ) respectively correspond to the error locator polynomial and the erase locator polynomial. The lb cycle shows the general processing steps for errors and erasure correction. In addition to receiving R (x), the indication calculator 20 also receives erasure data, and generates an indication polynomial S (x) and a Forney indication polynomial T (x). The bond equation solver 22 processes S (x) and T (χ) to generate an error and erase the evaluation polynomial Ω (χ), and an error and erase locator polynomial σ (χ) β, an error and erase locator polynomial Turn in the Chien searcher 24 to determine the wrong character 7 This paper size is applicable to China National Standard (CNS) A4 size (210X 297 mm) CII (Please read the note on the back side of the page and fill in this page)

• tT Liye Central Bureau of Standards and Quarantine Bureau Shellfish Consumer Cooperatives India India Central Government sample rate for Shellfish Consumers Cooperatives India A7 B7, V. The location where the invention description (4) number occurred, and the error and erasure evaluator Both the polynomial and the error and erase position are input to an error and erase value evaluator to generate the error and erase value. Techniques often used to solve bond equations include Berlekamp-Massey algorithms, Euclidean algorithms, and Continuous-fraction algorithms. Compared to the other two algorithms, the Berlekamp-Massey algorithm generally considers its hardware complexity to be minimal. ° A detailed description of the Berlekamp-Massey algorithm can be found in Article 7 of the Berlekamp Reference Citation, and published by J. Massey.

Synthesis and BCH Decodingy IEEE Trans, on Information Theory, IT-15: 122-127, 1969 ". An inversionless Berlekamp-Massey algorithm was proposed by Burton to eliminate high-cost Finite-field inverters (FFIs) β. For details, see HO Burton's "Inversionless Decoding of Binary BCH" Codes, IEEE Trans, on Information Theroy, IT-17: 464-466, 1971 "e. The conventional technology uses the traditional Berlekamp-Massey algorithm to calculate the error locator polynomial and the error evaluation polynomial, and as a design circuit Fundamentals. However, each algorithm requires a large number of finite field multipliers (FFMs), and possibly a finite field inverter FFI. Each FFM and FFI is converted into a hard cross-line circuit and implemented in Ji'ao Road. Therefore, the purpose of this invention is to derive an efficient polynomial solution and reduce the size (complexity) of the hard-crossed circuit when implementing the algorithm. The number of FFM and FFI is basically to change the paper size. Applicable to China Standards (CNS) A4 size (210X297 cm) (please read the precautions of κΓ before filling this page)

A7 B7 V. Invention Description (5) The function of the number t, the variable t is a function of (N-K) / 2. Table 1 shows the various algorithms and the corresponding numbers of FFM and FFI, when t is equal to 8. Reference algorithm FFM's (function of t) Number of FFM's Number of FFI's Berlekamp 3t 24 1 Liu 2t-l 17 1 Oh 2t 16 1 Reed 3rt + i > 27 0 As shown in Table 1, only for error correction (non-error and erasure correction) 'implementing a traditional Beriekamp-Massey algorithm (BeriekamP US sleep patent number requires 3t or 24 FFM's and 1 FFI's »Liu in r One of the algorithms demonstrated in the paper "Architecture for VLSI Design of Reed-Solomon Decoders, IEEE Trans, on Computers, Vol. 33, No. 2, February 1984" requires 2t-1 or 17 FFM's and 1 FFI's. In US Patent No. 5,583,499, Oh et al. Disclosed that a circuit requires 2t or 16 FFM's and 1 FFI's. On the other hand, the algorithm disclosed by Reed et al. Does not require inversion, so it is quite complicated The FFI of the degree is not required. The above algorithm is disclosed in "KL57 〇 / vers e-Free Berlekamp-Massey Algorithm, Reed, Shin, and Truong, IEE Proceedings-E, Vol. 138, No. 5, September 1991". However, even Reed's algorithm FI? I must be used, but it requires a large number of FFM's, as high as 3 (t + l) or 27. If applied to errors and erasure correction, the number of FFM's required will be higher, usually errors Revision status 9 (please read the items on the back-before filling this page)--ο. This paper size is applicable to the Chinese National Slope (CNS) A4 specification (210X297 mm) A7 B7 «The sample rate in the Ministry of Economic Affairs Xibeigong Consumer Cooperative Co., Ltd. Printed oxygen 5. Doubled under the description of invention (6). Therefore, when implementing the algorithm, a non-inversion method and its device without using FFIs and minimizing the number of FFMs are As expected, one object of the present invention is to provide a method and apparatus for solving a polynomial of a key equation when decoding a codeword. Another object of the present invention is to provide a method based on Berlekamp · Massey calculus. Method and its device, which can be implemented with the least hard lines. Another object of the present invention is to provide a method and a device for solving the polynomial of the healthy equation, which will not reduce the overall decoding speed of the decoder. In other words, in a better implementation In the example, a method for calculating a suspected locator polynomial and a false evaluator polynomial in an incorrectly-corrected brown solution to the key equation solution step formula is disclosed, and a polynomial is generated through some intermediate steps of the above method The above intermediate steps can be funded by the minimum number of hard-wired lines. The number of intermediate steps requires a corresponding number of operation time periods to complete the operation of the polynomial. However, depending on the selected (N, K) ma, the number of calculation time periods required to calculate the polynomial will be within the range of the time required to calculate up-stream data. In particular, it is efficient. Cut a small number of finite field multipliers (FFM's) without using a finite field inverter (FH's). The better method for calculating the error locator polynomial and the error evaluation polynomial is also revealed (please read the precautions on the back first) (Fill in this page again) AIV. Order CI. The paper size is used in China National Standards (CNS) 8 4 size (210X297 mm) Central Government Standards Bureau of the Ministry of Economic Affairs Consumer Cooperatives A7 B7 '' V. Invention Explanation (7). Using these new methods, implement the method derived from the Berlekamp-Massey algorithm without inversion. A structure with area efficiency (saving circuit area) using only 3 FFM's without FFI's is also disclosed. This method and architecture can be widely applied to various RS and BCH codes with appropriate coding length. The present invention provides a method and device. Another advantage of the present invention is to provide a method and device. Based on the Berlekamp-Massey algorithm, the number of hard II lines can be minimized. --------- o ^ .-- (please read the $ item on the back before filling in this page)

Another advantage of the present invention is to provide a method and device for solving the above-mentioned and other features and advantages of the present invention. After detailed explanation will be more obvious and easy. Brief description of the formula: The first circle shows the processing circle when the decoder has the error correction function code; the first circle shows the processing block diagram when the decoder has errors and the correction function code is erased; and the second circle displays one A preferred embodiment with three FFM architectures to implement the key equation solver of the present invention. Explanation of symbols: 10, 20 ~ Symptom calculator; 11 The size of the paper is applicable to the Chinese National Standard (CNS) A4 (210X297 mm) A7 B7 _ 5. Description of the invention (8) 12, 22 ~ Key equation solver 14'24 ~ Chien searcher; 16 ~ error evaluator; 26 ~ error and erasure value evaluator; 30 ~ input end; 32, 46, 48 ~ finite field multiplier; 34,50 ~ finite field addition 36, 40, 44, 54 ~ temporary register; 38, 52 ~ output; 42, 58 ~ multiplexer; 56, 60 ~ buffer; 62 ~ controller; and CLK1, CLK2, CLK3 ~ clock SIGNAL》 Comparative Example: Referring to the symbols used here, no symbols such as Ω and σ refer to the original Berlekamp-Massey algorithm (with inversion), and symbols with " such as cold, A left, Λτ ; Refers to the non-reverse algorithm. The inversion Berlekamp-Massey algorithm of the known technology is a 2t step (please read the precautions of back-xenon before filling this page)

， 1T AVT. «Repeated algorithm for the printing and cracking algorithm of the Peking Consumer Cooperative of the Ministry of Economic Affairs of the Ministry of Standards of China, as shown below: Initial conditions: π-υ = ο; (line 1) Λ δ = 1; (line 2) Cold Μ -1) ⑻ = ^-1) ⑻ = Λ (χ); (line 3) = 7 > +1 backup 浐 _1 > + 7 > minute -1) +. ·. · (Line 4) 12 sheets The scale is applicable to China's national standard rate (CNS) A4 specification (210 × 297 mm) A7 B7 V. Description of the invention (9) = Tp + iA〇 + Γ /, Λι + ·.. + ΆΚΡ for / = /? ToN-K- 1 Cold (,) ⑻ = ⑻. 今 (卜 1 巾) + 幺 (〇x 令 (卜 υ (χ) (Line 5) (Line 6) (Line 7) 4- * ((, + |) = r, + 2W0 + +… + Γ «-», + Ρ + 2 points d (line 8) If Α (<) = 0ογ2 £) (μ) 2 / + 1 (Line 9) DV) = Do-t). ^ ΊΟ (χ) = XT ^-· >(*); (line 10) else (line H) Di0 = i + \-Dv ~ l) > < 〇ί ^ (0 (*) = ff (< _,) (^); (line 12) where p is the number of erasure, between 〇 < PS NK; Λ〇ο = Π ( 1 + α Q, Λ is erasure set; Tj's is the coefficient of the Forney symptom polynomial T (x), where T (x) = Λ (χ) 5 (χ) mod xN K; 0W is the ith Wrong steps And erase the locator polynomial, the highest degree of the polynomial is Vi + p; Miao 4 is cold (a coefficient of 0W; (< > is the discrepancy value of step 1), "is the discrepancy value generated previously, r (<) (*) is the auxiliary polynomial, and D⑴ is the auxiliary variable. Here, the algorithm provides corrections for errors and erasures. • If there is no erasure, then P = 〇 'T (X) = S ( x), and cold-1⑻ = 卜 〇〇 = 1, and the algorithm is simplified to a simpler form. β The new error and erasure locator polynomial 所得 obtained from the non-inversion Berlekamp-Massey algorithm can be used to find The same recording error position as σ (χ) found by the original Berlekamp-Massey algorithm. As shown in the above, the i-th step without inversion Berlekamp-Massey algorithm contains the following two sets of equations (the above line 7 and 7) 8): σ (,) (χ) = ί · ^ 〇-ΐ) (*) + * r (, · (*) (equ. 1) 13 This paper uses China National Standard (CNS) A4 (210X297 public money) {Please read «Note $ on the back side before filling in this page） C.-* C. Duoyang A7 ___ B7, the co-operation between shellfish and consumer goods of the Central Biaofeng Bureau of the Ministry of Economic Affairs, V. Description of the invention (10 )厶 < < + ”=: Γ < + 2 points P + Γί + 1 point V) +… ^ + 2 (eqa2) In the present invention, we propose the following calculus definition: σ (Γ °» for j = Q (equ. 3a) σψ = δ σ) ι'ϋ + for \ ^ j < .Vi + P (equ. 3b) Left (') = kS!]. l + p + Γ, _ v,., + P +1 σ ry. \ VP. For j = 0 (eqa4a) Δ (J +,) = ^ 1-^ + for y = i ^ 7 ^ v / + ^ (equ. ^ B) where 彡 y), i is The coefficient of 彡 ⑺⑶, and the township correction 0 + correction 0 χ + ... + yin ", is cut to the highest degree without w (jc) polynomial, and the point is a coefficient of f (<) 〇〇. Then V + u · ί is the partial result of calculating λ (< + ”.” By the above definitions of 浐 and Α (广 ", in each calculation time period, only 2 FFMs, and only one FFM is required in the calculation (/ < +1). Using this method, only three FFMs are required in a calculation time period. By decomposing the original equations (equ. 1 and 2 ) Becomes a sequence of smaller calculation amount (equ. 3a, 3b, 4a, and 4b), which can quickly and drastically reduce the number of necessary FFMs. However, when calculating each individual value in any calculation time period, 'in_ There may be a data dependency between the shirt and the jacket. Table 2 shows the data dependency of this decomposition algorithm: (please read M-note on the back of the first page before filling in this page)

Cycle V ♦ ”σ ^ {χ) j = 〇j = lj = 2 ^ (〇 = ^ (v ?., + P + Γ, _ V,., + Χ, +! Σ ί;:, * > p ^ V +,) = Tt, 2 ^^ 〇 species) = "· fishing -〇 + milk about -ι > σψ = ί · σ§- ° + j = Vi + ^ V (: y = ^ v,; y Η · T ,. „_p + 3σ% p., • Sickle · Table 2 14 This paper size is free to use Chinese National Standard (CNS) 8-4 (210X29 * 7 mm) Industrial and Consumer Cooperatives Seal A7 B7 > V. Description of the Invention (11) As shown in Table 2, calculating the acupoints at the calculation time period j "requires (^ 仏 和 的 值 ', which has been calculated at the calculation time period j_l. Same as The calculation of calculations in the calculation time period j requires jealousy and Q--0, which have been calculated separately in the calculation time period 0 and (i-1) steps. Appendix A shows that using the algorithm of the preferred embodiment generates an error And erasing the evaluator polynomial and the error and erasing the locator polynomial. Using the above-mentioned decomposition algorithm, it is possible to use a 3_FFM non-inversion Berlekamp-Massey algorithm as a key equation solver, and such as As shown in lap 2. FFM 32, a first finite field adder 34 (FFA) 'and a register 36 are used to calculate the discrepancy value. In the jth operation time period operation of the i-th step, the FFM 32 receives the Forney symptom value Ti.j + 3 as the first input and received from the (jq) coefficient < ^ 2 as the second input. The FFA 34 and the register 36 accumulate the multiplication result β. With regard to the output 38, the error and erasure evaluator When the calculation of the coefficient of the polynomial valley is completed, the value of the coefficient will be provided to the output terminal 38. FFMs 46 and 48, and FFA 50 calculate the coefficient of the error and erase the locator polynomial & W. FFM 46 receives the non-compliance value A⑴ as One round in, the other input is "the multiplier 58 and the buffer 60 allow the selection and storage of 0. The buffers 56 and 60 store the coefficient 'multiplier 58' of the unit X (H) obtained in the previous step. Select the new value of 衧 -〇. FfM 48 receives one round of input and another input. The multiplier 42 and the register 44 consider the selection and storage. The outputs from FFMs 46 and 48 are added by FFA to generate the cape- ». Shirts are also stored in register 54 and fed back to buffer 56 and FFM 32. If A < 0 = 0 or 2Z ^-" 2, · + Ι, then India = right -1 & 15 U Zhang ^^ using Chinese S family standard (CNS) A4〇 ^ (21GX297 mm) --- ~ -------- JQ --- --- ir ------ Ό (Please read the notes on the back before filling out this page) «Jibuzhong * standard rate 扃 贝 工 消 # Cooperative Society Printing A7 ______B7 V. Description of Invention (12) Township Remain the same; otherwise, let p = & (丨 -〇 and》 = (00) register is used here as a delay element, the different clock signals CLK1, CLK2, and CLK3 generated by the controller 62 control. The register 44 updates its internal value during the first operation time period of each step. The internal value of the register 36 returns to zero during the second operation time period of each step. After 2t steps, the output value & can be obtained from the output terminal 52. This structure can be used for error correction or error and erasure correction. Compared with the previously proposed architecture, it needs 4t to 6t FFM's to implement correction and erasure correction or 2t to 3t FFM, s to implement error correction. The preferred embodiment of the present invention can greatly reduce the hard艟 Complexity to only 3 FFM's❶ However, in order to complete the i-th step algorithm, the architecture of the preferred embodiment requires Vi + p + 1 computation time periods, but the architecture of the conventional technology requires only 2 to 3 Operation time period. With the architecture of the present invention, the additional time required to generate data does not slow down the system's overall processing speed. One of the reasons is that the conventional technology architecture does not synchronize the use of time and time. It is difficult to know that the calculation results of the technology at any level can be obtained quickly, but in order for any data to be received and processed, it is necessary to wait for the results obtained by the upstream steps. In addition, the method and apparatus of the present invention minimize the hard Λ by using the same line as the calculation line (X). The traditional way of calculating errors and erasing the evaluator polynomial Ω〇〇 is to calculate σ (ΛΓ) in parallel. Using Berlekamp-Massey algorithm, this program includes one of the calculations of Ω (〇 (χ) 2t step inverse ft algorithm. However, if the highest 16 paper sizes are applicable to the Chinese National Standard (CNS) A4 specification (210X297 mm) (please read the notes on the back *. And then fill out this page) Order the Ministry of Economic Affairs The Central Government's Consumer Cooperation Du Yin * A7 B7 V. Description of the Invention (13) σ (χ) for V + P has been obtained first, then it can be obtained from the bond equation and Newton's identity: Ω ( Λ:) = θ (χ) σ (χ) ηκχΙ ^ _Λ:

Q = iS + i〇ir + -... 4i5q; i = Ql, ..., v + / > -L That is, after σ (χ) is obtained, the calculation of Ω〇〇 can be directly and more effectiveness. As demonstrated by Reed et al., The use of inversion-free Berlekamp-

Massey algorithm, prepare (x) = CCT (x); Therefore, by direct calculation, the following results can be obtained: ό (x) = S (x) a (x) mod xN ~ K > = CQ (x ). Using the Forney algorithm, showing that cold (x) and 6⑻ can produce the same error and erase the correction value ^ 4m ^) = e /. 〇 mc & (pri) In addition, it can be seen that the calculation of "• And the calculation is similar. Therefore, after the cold w is obtained, the same hard frame used to calculate #w can be reconstructed to calculate 60c). It can be calculated as follows: 6 ^ = 5, +, 0-0, forj = 0. In particular, referring to the second circle, FFM 32, FFA 34, and register 36 are used for calculation. To find the j-th operation time period of the i-th coefficient, FFM32 receives the symptom value Si # as an input, and the j-th coefficient center of the cold W as another input. The FFA 34 and the register 36 are used to accumulate multiplication results. When 々 (X) is obtained after 2t steps, the coefficient 孑 / will be stored in the buffer 56. Input 17 via setting buffer 60 or register 40 This paper size applies Chinese National Standard (CNS) A4 specification (2 丨 0 · 〆297 mm) -------- ο ------ ix ------ C, (Please read the notes on the back and fill in this page again) A7 ______B7 V. Description of the invention (14) The output is 0, and the output of register 44 is 1, and the output of buffer 56 is 1. Can be looped and fed back to the input of FFM 32. The directly calculated output (di) is available at output 38. When considering the use of the 3-FFM architecture of this embodiment, calculating the total number of calculation time periods required for the JC and the occupants, their impact on the overall performance of the system is of concern to us. From the iterative algorithm, it is suspected that the highest degree of W⑻ will increase by 1β at most during each iteration. Therefore, the equation v ^ vu 丨 +1 is used to determine the upper limit of Vj + p. The results of (1) error correction, and (2) error and erasure correction results are shown below. If it is only necessary to correct the error, then 2t < N-K, the calculation of A (0) only needs a calculation time period, and V, Q ·, / 扬 0U and νβί, / οτΉΚΜ » The number of calculation time encounters is: 21-1 f-1 2 < -1 ^ 1 Σ (ν < +1) $ Σ (ί + 1) + Σ (ί + 1) = 4ί2 + Yan ^. / * 〇 / «〇 /« # 2 2 The number of calculation time periods required to calculate ά⑻ is: = τί2 + -τί ° Therefore, the total number of calculation time periods required is less than 2t2 + 2t + l If both are corrected, then 2t + p < == NK, calculate the initial Δ < Λ shall be p + 1 operation time period, and ν, ^ ρ + i, for 0 ^ i < t »v, ^ p + tf for ⑸ < 2i. The number of time periods required to calculate 彡 ⑻ is: 1) = 1 ^ + (2 /, + 2) / 2i-l r-1 2i-l Σ (ν, + ι) < Σ (ρ + ι +1) + ΐ > + ί + i «0 < * 0 The number of calculation time periods required to calculate ά⑻ is: 18 This paper size applies to China National Standards (CNS) A4 parlance < 2 丨 OX297 mm) Printed by the Central Laboratories of the Ministry of Economic Affairs and Consumer Cooperatives «. Λ 7 Β7 V. Description of Invention (15) Therefore, the total number of calculation time periods required is less than 2t2 + (3p + 2) t + (l / 2) p (p + 1) + ρ + 1 »Because t and p are integers, under the constraint of 2t + p < = NK, there is no (t, p) closed-pin formula that will maximize the total number of operation time periods. (Closed-form formula). Instead, for various (N, K) RS codes, the N-K encoding length ranges from 4 to 16, and the total number required for the calculation time period has been calculated and shown in Table 3. If N is greater than the required number of operation time periods, the method and device of the present invention can therefore be used to reduce hardware complexity while still maintaining a tidy decoding speed. , NK tp cycles 4 2 雠 13 4 1 2 16 6 3-25 6 1 4 31 8 4-41 8 2 4 51 10 5-61 10 2 6 76 12 6 • 85 12 3 6 106 14 7-113 14 3 8 141 16 8-145 16 4 8 181 Table 3. In communication and storage systems, BCH and RS codes have multiple applications, all of which can benefit from the method and device of the present invention. For example, digital versatile disks (DVDs) use RS to generate codes, which are (182,172) in the column direction and (208,192) in the block direction; 19 -------- ο-- ---- II ------ C (Please read the precautions on the back before filling this page) This paper size is applicable to the Chinese national standard (CNS specification (210X297 mm) A7 B7 V. Description of the invention (16 ) Digital TV broadcasting uses (204,188) RS code; CD-ROM uses multiple smaller RS codes, including (32,28) and (28,24); In wireless communication, AMPS cellular mobile phone system uses (40 , 28) and (48,36) are binary BCH codes, which are reduced codes of (63,51) codes. (63,51) codes of 2 errors (NK = 12, m = 6) can be corrected, Requires less than 12 computing time periods (t = 2, the first column of Table III). All applications such as these and other aspects can benefit from the method and device of the present invention. Although the present invention has been better implemented The example is disclosed as above, but it is not intended to limit the present invention. Any person familiar with the art, without departing from the spirit and scope of the present invention, proposes modifications and retouching. Within the scope of protection of the present invention, and the scope of protection of the present invention shall be determined by the scope of the attached patent application "-------- Q ------ 1T ------ ο, (Please read the notes on the back * · before you fill out this page) The paper size printed by the Central Standards Bureau of the Ministry of Economic Affairs, Shellfish Consumer Cooperative, is applicable to the Chinese National Standard (CNS) A4 specification (2 丨 0X297 mm) Standards Bureau Λ Labor ¾ Fees Cooperatives Seal «. A7 B7 V. Description of Invention (17)

Appendix A Berlekamp-Massev Decomposition and Wing Decomposition Method Part I: (Calculate Cold Heading)

Dip ~ l) = 〇, »= l; / * initial conditions * / ^ = Tp ^ K. * TPK ^ * Tthp for ϊ = /? To N-K-1 begin / * external loop start * / bismuth, In r) / if i! = P in ⑴ = M? .I + p + r BU ,, ... AH; Δ (〇 +,) = 0 for j = 1 to v, + p begin / * internal forced circle starts * / 々 卜 沾 + ⑽⑽ Α〇 + »Α« + ·) ^ ^ Λ < 〇 ** > = Δ / -ι + Γι-/ + 3 tfy-i end loop / * End internal loop * /

If Δί0 = 0 ογ2 £ > (Μ) ϋ + 1 / > ω = £ ^ W) = xf) ⑻; else

2) (0 = ί + 1-do Bu ", including = 幺., The same. (Jc) = >(x); end loop / * End the external loop V • min + do r ~ ·· + cv ”21 This paper size is applicable to the national standard rate (CNS) Α4 specification (210 × 297 mm) -------- Q ------ 1T ------ C, (Please read the back · Please fill in this page and fill in this page) A7 B7 V. The invention description (is) will be stored 〇 + dl JC +… + θν + / >. Part II: (Calculate 1) Ω ^ 0) = Si σο for i = 1 to υ + ρ-l begin tL, = for j = 1 to i begin end loop end loop ft ⑻ = d +. · ++ 〇- 丨 Δ Λ. + Al + ... + Av + pl jpV + P 1 e-(Please read the note of v # and fill in this page again 4) Order-C1. Printed copy of the shellfish consumer cooperative of the Central Sample Rate Bureau of the Ministry of Economic Affairs Paper size applies to China National Standards (CNS) A4 specifications (210X297 mm)

Claims (1)

il C8 ---- ^ _ Ξ! __: __ VI. Patent application difficulties · A method 'used to calculate the key equation polynomial in decoding the received code space after error correction coding processing, the above key equation polynomial includes an error A locator polynomial to indicate 0 or more error positions in the received codeword 'and an error evaluator polynomial to indicate 0 or more recorded error values relative to the above 0 or more error positions. The above method includes the following steps: (a) receiving a plurality of first signs, which indicate the error pattern of the codeword received after the error correction encoding process; (b) generating an error locator polynomial coefficient, which is a previously generated non-conformance value, which has been generated Function of the error locator polynomial coefficient, a generated mismatch value, and a generated auxiliary polynomial coefficient function; (c) generating a local mismatch value, which is a local mismatch value that has been generated, a first sign described above The first special value, and a function of the polynomial coefficients of the generated error locator; (d) repeating steps (b) and (c) to generate a plurality of the above error locators Polynomial coefficient, which represents an error locator polynomial; (e) Generate a local error evaluation polynomial coefficient, which is a generated local error evaluation polynomial coefficient, a second special value of the first sign, and the error Function of locator polynomial coefficients; (f) Repeat step (e) to produce complex numbers. The above-mentioned erroneous evaluation polynomial coefficients represent a erroneous evaluation polynomial. 2. The method according to item 1 of the patent application park, wherein when generating the j-th coefficient of the i-th step erroneous locator polynomial, the above-mentioned step b 23 uses the Chinese national standard (CNS > Λ4 specification (210X297mm) (please read the note of Φ before reading) όΜ. Order the central standard of the Ministry of Economic Affairs 舅 Industrial Consumer Cooperative Cooperative India and the Central Government Standards Bureau of the Ministry of Economic Affairs 舅 Industrial Economic Cooperative Cooperatives The above-mentioned generated non-compliance value in the patent application factor is the i-th generated non-compliance value β 3. The method as described in item 2 of the scope of patent application, in which an i-th step of the error locator polynomial is generated. In the case of coefficients, the above-mentioned error locator polynomial coefficient generated in the above step b is the j-th generated coefficient of the error locator polynomial in step (il). 4. The method described in item 3 of the patent application park, When the j-th coefficient of the i-th step error locator polynomial is generated, the auxiliary polynomial generated in the above step b is the (jl) -th generated coefficient of the (M) -step auxiliary polynomial. Paragraph 1 Method, wherein when a j-th partial result of the (i + 1) th non-conforming value is generated, the above-mentioned generated non-conformity value in the step e above is the (i + 1) -th non-conforming value of the ^^ generated local part Result. 6. The method described in claim 5 of the patent specification, wherein when a j-th partial result of (i + 1) discordant values is generated, one of the first signs in the above step c is first The special value is the (h + 3) value of the above-mentioned first symptom. 7. The method as described in item 6 of the patent application park, wherein when a j-th partial result of the (i + 1) -th non-conforming value is generated In the above step, the above-mentioned generated error locator polynomial coefficient in c is the (j-1) th coefficient of the error locator polynomial recorded in step i. 8. The method as described in item 1 of the patent application, where When generating the j-th local result of the polynomial coefficient of the i-th recorded error evaluator, the polynomial coefficient of the error locator generated in the above step 骒 e is wrong. A4 specification (210X297 mm) (please read the note of K first and then write it down) C —------------ -"Finch ------ ^ ------ The rate of bidding in the Zhongji Department is the consumption of shellfish, including the seal of the company«. A8? S -------------- D8___ The j-th coefficient of the polynomial of the patent application locator. 9. The method described in item 8 of the patent application group, in which when generating the j-th local result of the i-th polynomial coefficient of the error evaluator The above step refers to the local result of the above-mentioned error evaluator polynomial in e as the (j-1) th local result of the polynomial coefficient of the error evaluator. 10. As described in item 9 of the patent application park, When the j-th local result of the polynomial coefficient of the i-th error evaluator is produced, the second special value of the first sign in the above step 骒 e is the (i-j + 1) value of the first sign of death . 11. A method for calculating polynomials of plating equations in decoding received codewords that have undergone error correction and erasure correction coding, the key equation polynomials include an error and erasure locator polynomial to indicate 0 or Multiple errors and erasure positions, one error and erasure. The evaluator polynomial uses α to indicate 0 or more error and erasure values. With respect to the above zero or more error and erasure positions, the above method includes the following steps: ( a) receiving a plurality of the above-mentioned first micro-megabits, which indicate the received error patterns of the characters that have undergone the error correction coding process, and receiving a plurality of Forney levies; (b) generating an error and erasing the locator polynomial coefficients, It is a function that has previously generated a discordant value, one that has generated errors and erased the locator polynomial coefficients, one that has generated the discordant values, and one auxiliary polynomial coefficient that has been generated; (c) produces a local disagreement value, which It is a local discrepancy value that has been generated, one of the above Forney symptoms, and an error that has been generated and erased. 25 Zhang ΛΑ is applicable to China National Standard (CNS) eight specifications 2ι〇 × 297 mm) --- ---------- tr ------ 9 (Please read the note f on the back and then f on this page) M but the central slope * ^ The labor cost includes the print of the company «. A8 B8 C8 D8, a function of the polynomial coefficients of the patent application Fan Locator polynomial; (d) Repeat steps (b) and (c) to generate a plurality of the above errors and erase the Locator polynomial Coefficient, which represents an error and erasure locator polynomial; (e) generating a local error and erasure evaluation polynomial coefficient, which is a local error and erasure evaluation polynomial coefficient that has been generated; First, and the function of the above error and erasure polynomial coefficient of the locator; (f) Repeat step 骅 (e) to generate a plurality of the above error and erasure evaluation polynomial coefficients, which represents an error and erasure evaluation polynomial . 12. A device for calculating a key equation polynomial in decoding an error-corrected encoding codeword. The above-mentioned robust equation polynomial includes an error locator polynomial and a false evaluator polynomial. The device includes: (a) — The first sub-hail road is used to receive an indication that it indicates an error pattern of the iodine-treated codeword received after being incorrectly corrected, a local discrepancy value that has been generated, and a polynomial coefficient of the mislocator in order to generate A local mismatch value; and (b) —the second sub-circuit receives a previously generated mismatch value, a generated mismatch value, a generated error locator polynomial coefficient, and a generated and generated auxiliary polynomial coefficient, In order to generate an error locator polynomial coefficient. 13. The device as described in claim 12 of the patent specification, wherein the first sub-circuit includes a first multiplier, a first adder, and a paper scale using China's national operating rate (0?> ; 8 4 specifications (210 > < 297 mm) Printed A8 B8 C8 D8 patent application Fangu first storage element printed by the central government's frank consumer cooperatives 14. The device described in item 13 of the patent application, wherein the first sub-circuit includes a first multiplication A multiplier, a first adder, and a first storage element. The multiplier receives the multiple coefficients of the error locator and the symptoms as inputs to generate a round of the first adder. The first adder generates an output. To a first output terminal and the first storage element, the storage element is returned to the first adder for performing a totalizing operation. 15. The device according to item 12 of the scope of patent application, wherein the second sub-circuit includes two multipliers'-adders, and a plurality of storage elements, a complex multiplier, and a plurality of buffers. 16. The device according to item 15 of the scope of patent application, wherein the second sub-circuit includes a second storage element for receiving and selectively storing a first signal and providing the stored first signal to a first Two multipliers and a first multiplexer. The second multiplier receives a first buffered signal as a second input to generate an output to a second adder. The first multiplexer receives a wide feedback signal. As a second input to generate a signal to a third storage element, the third storage element provides a storage signal to a third multiplier as an input, and the third multiplier receives a second buffered signal as a second Turn-in to generate a round-out signal to a fourth storage element and a second output terminal. The fourth storage element provides storage information to the first sub-channel and a first buffer, and the first buffer generates The second buffer signal is sent to the third multiplier and a second multiplexer, and the second multiplexer generates a selection signal to a 27th good note. The scale is applicable to China National Standard (CNS) A4 specification (2 丨0X297 Mm) -------- II (# ¾Read the back West < Note f, then fill in this purchase) Order · M Ministry of Central sample rate f Labor clearance fee included as a company seal *. Il C8- ---------- 、 Applicable patent scope: a buffer to provide the first buffered signal to the second multiplier 'to provide the first buffered signal to the second multiplier and the second multiplexer Device. 17. The device according to item 12 of the scope of patent application, wherein the first sub-circuit generates the local discrepancy value in a first period, and generates a local evaluator polynomial coefficient in a second period. 18. The device according to item 17 of the patent application, wherein the first sub-circuit receives a symptom, a locally generated value of an error evaluator polynomial coefficient, and an error localization during a second time encounter period. The polynomial coefficients are used to generate a local result of the polynomial coefficients of the error evaluator, and the second sub-circuit can allow the polynomial coefficients of the error locator to pass to the first circuit beside the first sub-circuit for processing. The device according to item 3 of the patent application park, wherein the multiplier and the adder are finite unit multipliers and adders. 20. The device according to item 14 of the patent application park, wherein the complex multiplier and the complex adder are finite field multipliers and adders. 21. The device according to item 15 of the patent application park, wherein the complex multiplier and the complex adder are finite field multipliers and adders. 22. The device according to item 16 of the patent application park, wherein the complex multiplier and the complex adder are finite field multipliers and adders. 23. — 秾 device for correcting errors after decoding and erasing. Computation of a healthy equation polynomial in the processed received codeword, the above key equation polynomial includes an error and erasure locator polynomial, and an error of 28 paper sizes, using the Chinese standard for home papers (CNS > A4 specification (210X297 mm) ] ~ '-------- C Pack—— < Please read the note of the back cloth first and then buy it in 4 copies.) Order the Central Standards Bureau of the Ministry of Economic Affairs 丨 Industrial Consumer Cooperatives Print It Apply for patent Fan Gu and The device for erasing the evaluator polynomial includes: (a) a first sub-circuit for receiving a Forney sign, a locally discrepant value that has been generated, and an error and erasing the locator polynomial coefficients to produce a local Non-conforming values; and (b) —the second sub-circuit receives a previously generated non-conforming value, a generated non-conforming value, a generated non-conforming value, a generated error and erases the locator polynomial coefficients, and Once produced Auxiliary polynomials in order to generate an anchor mislocator polynomial grandchild. 24. The device according to item 23 of the scope of patent application, wherein the first sub-circuit generates the local discrepancy value in the first period, and in the second period A local corrigendum evaluation polynomial coefficient is generated in the device. 25. The device as described in item 24 of the scope of patent application, wherein the first sub-circuit receives the first symptom value in a second period, an error and erasure evaluation. The local value of one of the polynomial coefficients of the localizer, and 巳 the error and erase the localizer polynomial coefficients to generate an error and erase the local value of the localizer polynomial coefficients, and the second sub-circuit can allow the above 锵The error and erasure polynomial coefficients of the locator are passed to the first circuit above the first sub-circuit for processing. 26. In a system for receiving codewords for deiodination and error correction coding processing, there is a sign The calculator receives the above codeword and generates a symptom, and the one-button equation solver receives the above symptom and generates a clamp error evaluator polynomial and a false position. Terms, a Chien searcher is used to receive the above error locator polynomial to generate an error position, and an error value evaluator is to receive the above error evaluator polynomial and the above 29 scales, using the Chinese National Standard (CNS) A4 specification (210X297 mm) (Please read the note of the fate of the fate first, and then f this page) Order- ^ l_i m 11 The Ministry of Economic Affairs of the People's Republic of China won the bid of the Hongbei Gonggong House for Du 4-pack A8 B8 C8 D8 An error value is generated relative to the recorded error position. An improved method provided to the robust equation solver is to indicate 0 or more error positions in the received codeword in order to generate the error locator polynomial and generate the above-mentioned calculation. The valuer polynomial indicates 0 or more error values with respect to the above 0 or more error positions, including the steps of: (a) receiving a plurality of first signs, which represent the received error-corrected encoding processing codewords; Error pattern; (b) generating an error locator polynomial coefficient, which is a previously generated mismatch value, a generated error locator polynomial coefficient, a generated mismatch value, and a generated auxiliary polynomial coefficient function ; (C) generate a local discrepancy value, which is a generated local discrepancy value, a first special value of the above-mentioned first symptom, and a function of the polynomial coefficient of the generated error locator (& 0 Repeat steps (b) and (C) to generate a plurality of the above-mentioned error locator polynomial coefficients, representing an error locator polynomial; (e) generate a local unitary error evaluator polynomial coefficient, which is a The local error evaluator polynomial coefficients generated, a second special value of the above-mentioned first symptom, and the function of the above-mentioned error locator polynomial coefficient; (f) step (e) is repeated to generate a plurality of the above-mentioned false error evaluations The polynomial coefficient of the determinant represents an error evaluator polynomial. 27. The system described in item 26 of the patent application park, wherein when generating an i-th step of the j-th coefficient of the error locator polynomial, the above step b The discrepancy that has been generated is the i-th discrepancy that has been generated. 28. The system described in item 27 of the patent application, in which 30 winter papers are produced on a perpetual scale using the Chinese national standard (CNS > A4 wash grid) (210X297 mm) C ------ ^ ------ C (please read the "$" note of ^ -face * before filling out this page) A8 B8 C8 D8 S84425 When the j-th coefficient of the polynomial of the locator in the i-th step is wrong, The error polynomial coefficient of the raw locator is the j-th generated coefficient of the error locator polynomial in step ⑴ ^. -------- Q! (This page) 29. The system described in item 28 of the patent application form, wherein when generating the j-th coefficient of the i-th step error locator polynomial, the above-mentioned auxiliary polynomial generated in step b is the step-assistance The G-1) th polynomial generated coefficient. 30. The system as described in item 26 of the scope of patent application, wherein when generating a j-th local value of the (i + 1) -th non-compliance value, the above-mentioned non-compliance value generated in step c above is (i + I) ) Local results of the (j_l) th inconsistent values. 31. The system described in item 30 of the scope of patent application, wherein when a j-th local value of the (i + 1) -th non-conforming value is generated, the first special value of the first sign in the step c is the above The (i · j + 3) value of the first sign. 32. The system described in item 31 of the scope of patent application, wherein when the jth local value of (i + 1) non-conforming values is generated, the above The above-mentioned generated error locator polynomial coefficient in step c is the (j-1) th coefficient of the error locator polynomial in step i. 33. The system described in item 26 of the patent application, wherein when the j-th local value result of the i-th error evaluator polynomial coefficient is generated, the above-mentioned error locator polynomial coefficient generated in the above step 骅 e Is the j-th coefficient of the error locator polynomial. 34. The system described in item 33 of the patent application, which produced 31 paper sizes using China_Jiaquan (CNS) A4 specifications (210X297) by M Ji Yet Central # 率 局 贝 工 * FE Cooperative ί 384425 Af C8 ---- --------- D8___ * VI. When applying for the patent, the first local result of the i-th cone cone evaluator polynomial coefficient results in the above step e The local result of the error evaluator polynomial that has been generated is the (jd) th local result of the first error evaluator polynomial coefficient. 35. The system described in item 34 of the patent application park, wherein when the i-th local j-value result of the polynomial coefficient of the error evaluator is generated, the above-mentioned step is called the above-mentioned second special feature of the above-mentioned first sign The value is the (i-j + 1) -th value of the first sign described above. 36. In a system for decoding received mamma characters that have undergone error and erasure correction coding, the system has an indication calculator to receive the above codeword and an erasure set to produce an indication polynomial and a Forney indication polynomial A one-button equation solver generates an error and erasure evaluator polynomial and an error and erasure positioning polynomial by receiving the above symptoms and the above Forney symptom. A Chien searcher is used to receive the above errors and erase the locator polynomial. Error and erase position, and an error and erase value evaluator to receive the error and erase evaluator polynomial and the error and erase position to generate errors and erase relative to the error and erase position Dividing value, an improved method provided to the above-mentioned unitary equation solver is to indicate 0 or more errors and erasure positions in the received codeword in order to generate the above-mentioned error and erase the locator polynomial, and generate the above-mentioned evaluator Polynomials that indicate 0 or more phases that are surprised by the above 0 or more errors and erasure values and erasure values, including the steps (a) Receiving a plurality of first micro-megabits, which represents the received error type of the word that has undergone error correction coding processing, and a plurality of Forney sign 32 paper waves, using the Chinese national standard ♦ (CNS > Α4 specification ( 210X297mm > -------- 1, Poly-/ V (please note the $ item first, then fill out this page) Order S8 ^ 425 (b) Generate an error and erase locator polynomial coefficient, which is a previously generated mismatch value, an error and erase locator polynomial coefficient that has been generated 'a mismatch value that has been generated, and an auxiliary polynomial that has been generated Function of coefficients; (< 0 produces a local discordance value, which is one of the generated local discordance values' one of the above Forney signs, and a function that has generated errors and erased the polynomial coefficients of the locator; (d ) Repeat steps (b) and (c) to generate a plurality of the above-mentioned error and erase locator polynomial coefficients, which represent a recorded error and erase locator polynomial; (e) generate a local error and erase evaluator polynomial Coefficient, which is a function that has generated local errors and erased the evaluator polynomial coefficients, one of the above-mentioned first signs, and the function of the above errors and erased the polynomial coefficients of the locator; (f) repeat step (e ) To generate a plurality of the above-mentioned error and erasure evaluator polynomial coefficients, which represent an error and erasure evaluator polynomial. 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