JP2003168983A - Decoding circuit and decoding method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for decoding a cyclic code and a decoding circuit in which a coefficient calculation circuit of an error position multinomial equation Λ (Z) is the same, a repetition count of a calculation loop is always a fixed count, and a condition determination is small, in the method for decoding a cyclic code and the decoding circuit in which t-piece or less of error is correctable. <P>SOLUTION: With the arrangement, a calculation is made in a coefficient calculation circuit 16 of an error position multinomial equation by using a matrix-reduction algorithm in a matrix of (t+1) row and t column obtained by adding an additional row of 1 row and t column to a Hankel matrix (square matrix). Thus, the coefficient calculation circuit 16 of the error position multinomial equation is same, a repetition count of a calculation loop is always a fixed count (the error correctable number: t times), and a condition determination is small, so that an increase in a circuit size can be restricted. <P>COPYRIGHT: (C)2003,JPO

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a cyclic code (eg, Reed-Solomon code (RS code), BCH (Bose-Chaudhuri-Hocqenghem) code, etc.).
TECHNICAL FIELD The present invention relates to a decoding circuit and a decoding method for decoding a code sequence encoded by.

[0002]

2. Description of the Related Art A conventional decoding method will be described. Cyclic code encoding,
And the decoding can be formulated as a polynomial operation using the elements above the finite field GF (q) (= GF (p ⁱ )). If p is a prime number, n is a positive integer relatively prime to p, and m is the smallest positive integer i that satisfies the relationship p ⁱ ≡1 mod n, then the order n
The element α of exists in GF (p ^m ), and the generator polynomial

## EQU1 ## GF (q) (= GF (p, which has a sequence of 2t zeros of α ^b , α ^{b + 1} , ..., α ^{b + 2t-1.
i} )) is a cyclic code with a code length n and the number of information symbols is k (q-ary (n, k) cyclic code). However,

## EQU00002 ## q = ^p.sub.i , i | m (i divides m). This cyclic code has a capability defined by consecutive zeros, that is, a design distance (2
It is a code capable of up to t algebraic error corrections based on (t + 1). Having a series of zeros in the generator polynomial g (X) also has zeros that are conjugate with respect to these GF (p ⁱ ), but those that do not belong to consecutive zeros are not used.

Encoding of a q-ary (n, k) cyclic code is performed by calculating the following code polynomial c (X).

## EQU00003 ## c (X) = i (X) g (X) Although this encoding is performed, the following systematic encoding is generally performed because of the ease of extracting information symbols after decoding. .

## EQU00004 ## c (X) = i (X) ^X.sub.nk- r (X) where:

[Mathematical formula-see original document] i (X): polynomial r (X) having information symbol as a coefficient: polynomial r (X) = i (X) X ^nk mod g (X) g (X) having a parity check symbol as coefficient: Generator polynomial g (X) = LCM (m _b (X), m _{b + 1} (X), ..., M
_{b + 2t-1} (X)). However, m _j (X) is G of the element α ^j of GF (p ^m ).
It represents a polynomial having all conjugate elements of F (p ⁱ ) as zeros, that is, a minimum polynomial. In particular, since in the case of a Reed-Solomon code, conjugated source regarding alpha ^j of GF (p ⁱ⁾ is the alpha ^j itself,

[6] g (X) = (X- α b) (X-α b + 1) ... (X-
α ^{b + 2t-1} ).

Here, in order to explain the decoding, the Fourier transform and the inverse Fourier transform on the finite field will be described. Let p be a prime number. Now, for a positive integer n relatively prime to p,

## EQU00007 ## When m is the smallest positive integer i that satisfies the relationship of p ⁱ ≡1 mod n, an element α of the order n exists in GF (p ^m ). Where GF
(P ^m) sequence in the time domain of length n on

[Equation 8] And the frequency domain series

[Equation 9] think of. For each sequence, an undetermined element X in the time domain and an undetermined element Z in the frequency domain are used to give polynomial expressions of the following equations (1) and (2).

[Equation 10] In this specification, a sequence and a polynomial corresponding to the sequence may be used interchangeably. At this time,

[Equation 11] These two polynomials (sequences) are a Fourier transform pair over a finite field. The transform from the time domain to the frequency domain is called the Fourier transform, and the transform from the frequency domain to the time domain is called the inverse Fourier transform.

Decoding of a cyclic code is performed by a receiving polynomial.

## EQU12 ## This is an operation for estimating the code polynomial c (X) from y (X) = c (X) + e (X). However, the error polynomial e (X) is an error polynomial having an error symbol as a coefficient, the number of errors is ν (≦ t), and the error positions are j ₁ , j ₂ , ... When the error amount at E _j1 , e _j2 , ..., E _j ν is represented by the following equation (5) using the undetermined element X.

[Equation 13] Similarly, y (X) is also expressed by the following equation (6).

[Equation 14]

The decoding method can be roughly divided into two.
That is, (1) a method of comparing a sequence corresponding to the reception polynomial y (X) with a sequence corresponding to all the code polynomials, and using the code polynomial having the shortest Hamming distance as the estimated code polynomial (the maximum likelihood decoding method), (2) By calculating the Fourier transform on y (X) on the finite field described above, 2t continuous spectra in the frequency domain (this spectrum is generally called a syndrome) are calculated, and the 2t spectra (syndrome) are calculated. A method of obtaining an estimated error polynomial by using it, and subtracting the estimated error polynomial from the received polynomial to obtain a code polynomial (algebraic decoding method). Of these, the method (1) is used only when the number of code polynomials (the number of codewords) of the code is small in terms of the scale of the memory. Therefore, in general, the algebraic decoding method (2) is widely used. The prior art will be described below with respect to the algebraic decoding method which is most relevant to the present invention.

In the algebraic decoding method, decoding is performed in the following six steps. Step 1: Calculate 2t spectra (syndrome) from the reception polynomial y (X) by Fourier transform on a finite field. Step 2: If all syndromes are zero, it is determined that there is no error. Otherwise, go to step 3. Step 3: Obtain an error locator polynomial and an error evaluation polynomial from the syndrome. Step 4: Obtain the zero point of the error locator polynomial and estimate the error position. Step 5: Estimate the error amount corresponding to the error position from the error locator polynomial and the error evaluator polynomial to obtain the estimated error polynomial e
Calculate (X). Step 6: From the received polynomial y (X) to the estimated error polynomial e
(X) is subtracted and error correction is performed (estimated code polynomial is calculated).

The above steps are executed by the decoding circuit 50 shown in FIG. 6, for example. Decoding circuit 50 shown in FIG.
Is a syndrome calculation circuit 52, an error position polynomial coefficient calculation circuit 54, an error evaluation polynomial coefficient calculation circuit 56, an error position calculation circuit 58, an error amount calculation circuit 60, an error polynomial calculation circuit 62, an output circuit 64, and Delay circuit 66
Equipped with.

Calculation of the syndrome in step 1 is performed by using GF
Original α of (p ^m) GF to the receiving polynomial y (X) on the (p ^{m)
It is performed} by substituting ^b , α ^{b + 1} , ..., α ^{b + 2t-1} (that is, Fourier transform on the finite field GF (p ^m )). That is,

[Equation 15] Calculated by For i such that b ≦ i ≦ b + 2t−1, always

Since c (α ⁱ ) = 0,

Since S _i = y (α ⁱ ) = c (α ⁱ ) + e (α ⁱ ) = e (α ⁱ ), there is no error if all 2t syndromes are 0 in step 2, If any of them is not 0, it is judged that there is an error.

When there are non-zero 2t syndromes, 2t syndromes S _{i in the} algebraic decoding method.
Error correction is performed by estimating the error polynomial from. That is, after obtaining the syndrome, which is the symptom caused by the error polynomial e (X), from the reception polynomial y (X), the error polynomial e (X) with the smallest Hamming weight of the corresponding sequence is obtained using this syndrome. In step 3, an element α ^j1 , of GF (p ^m ) indicating the error positions j ₁ , j ₂ , ..., jν,
The coefficient of the error locator polynomial Λ (Z) of the following equation (7) having the inverse ^elements α ^-j1 , α ^-j2 , ..., α ^-j ν of α ^j2 , ..., α ^j ν as a zero point is calculated.

[Equation 18] However, ν ≦ t. Here, the coefficient Λ ₁ of Λ (Z),
Λ ₂ , ..., Λν are values that satisfy the following expression (8).

[Formula 19] As an algebraic decoding method for obtaining the coefficients Λ ₁ , Λ ₂ , ..., Λν satisfying the equation (8) in polynomial order, the Peterson method and the Berlekamp-Massie method are used.
-Massey method (BM method), and Euclid (Eucli
d) methods have been proposed, which are respectively O (t ³ ),
O (t ^2), and obtains an error position polynomial O the number of multiplications (t ²⁾ (calculation amount).

The coefficients Λ ₁ , Λ ₂ , of the error locator polynomial Λ (Z),
..., after Λν is determined, determine the zero point of the error locator polynomial in Step 4 to determine the error location j _i. In general,
Substituting α ^-i into the error locator polynomial Λ (Z) sequentially,

A method called a Chien search for searching α ^-i that satisfies Λ (α ^-i ) = 0 is performed.

In step 5, the error evaluator polynomial is generally used.

[Equation 21] (Forney) using Λ '(Z) obtained by formal differentiation (derivative) of the error locator polynomial Λ (Z)
The amount of error is calculated by an algorithm. However, S
(Z) is a syndrome polynomial whose syndrome is a coefficient

[Equation 22] Is. The error amount e _ji is

[Equation 23] (Forney algorithm).

In step 6,

C (X) = y (X) -e (X) Error correction by.

Here, the conventionally known algebraic decoding method and the problems in circuitizing the algebraic decoding method will be summarized.

The Peterson method and the Beerkamp method which are often used in obtaining error locator polynomials and error evaluator polynomials
The problems of the Massey method and the Euclidean method are described respectively. When the decoding and the decoding circuit are realized by the Peterson method with the amount of calculation O (t ³ ), it is necessary to prepare the coefficient calculation circuit of the error locator polynomial Λ (Z) corresponding to all the error numbers ν of t or less. Yes, the circuit scale is O
(T ^3), rather than taking the sum of 1 and O (t ³⁾ until t

[Equation 25] Therefore, it becomes O (t ⁴ ). When the decoding and the decoding circuit are realized by the Beerkamp-Massie method of which the calculation amount is O (t ² ), a plurality of condition judgments are included in one operation loop, which makes the control circuit complicated and increases the circuit scale. Similarly, when a decoding circuit is implemented by the Euclidean method with a computational complexity of O (t ² ), polynomial division is performed, and therefore it is necessary to determine the degree of the polynomial, which increases the complexity of processing. However, this not only causes an increase in the circuit scale, but also increases the control complexity. Further, the Euclidean method has a calculation amount of O (t ² ), but since the number of iterations of the operation loop is not constant, the circuit structure is not regular and circuitization is difficult.

When performing the above-described error correction, it is necessary to delay the reception polynomial by the calculation time until the estimated error polynomial that starts with the syndrome calculation is calculated. Generally, in a digital communication system and a digital storage system, the reception polynomial received by the reception side through the communication path or the storage medium is not one, but a plurality of reception polynomials are continuously received. In such a continuous digital communication and digital storage system, it is necessary to delay the reception polynomial by a fixed time and reduce the obtained estimation error polynomial. However, in the Beerkamp-Massie method and the Euclidean method, the time for obtaining the estimated error polynomial is not constant depending on the number of errors, so it is necessary to delay the estimated error polynomial with a delay time corresponding to the time required to obtain the estimated error polynomial. It was Furthermore, for this reason, it is necessary to delay the receiving polynomial by the maximum delay time required to obtain the estimated error polynomial.Thus, the throughput depends on the maximum delay time even though the calculation amount is suppressed by making conditional judgments. Is dominated. From the above, if the time until the estimated error polynomial is determined is constant and the condition for condition determination is small, the circuit becomes easy regardless of the number of errors.

In estimating the error position and the error amount, if the operations after the Chien search are simply parallelized, it is necessary to prepare n dividers at the stage of the Forney algorithm for obtaining the error amount. That is, even if the amount of calculation of the Forney algorithm is O (nt), the prepared circuit scale is O.
(N ² ). However, not all of the n dividers are always used, but only the maximum number t of error correctable dividers, t, is used. Further, even if the dividers are shared, a circuit for allocating the dividers according to the error position is required, so that the circuit scale is hardly improved.

Further, whether or not the error polynomial estimated to have occurred is decodable (decodable / impossible) is determined using the Peterson method by using the error locator polynomial Λ (Z) and the error evaluator polynomial Ω (Z ), The number of errors ν obtained by determining the regularity of the determinant, the error locator polynomial Λ (Z), the error evaluator polynomial Ω (Z), and the error locator polynomial Λ (Z ), And the number of zeros is used for the following three conditions. (1) ν ≧ 1 (where ν ≦ t) (2) deg (Λ (Z))> deg (Ω (Z)) (3) deg (Λ (Z)) = (GF of Λ (Z)
(Number of zeros on (p ^m )) Decoding is possible if all of the above three conditions are satisfied (when the number of errors estimated to have occurred is t or less and any one of the three conditions is not satisfied). Is undecodable (the number of errors estimated to have occurred is (t + 1) or more).

Similarly, using the Beerkamp-Massie method, the error locator polynomial Λ (Z) and the error evaluator polynomial Ω
When (Z) is obtained, the error locator polynomial Λ (Z), L
Using the register length L of the FSR and the number of zero points of the error locator polynomial Λ (Z) obtained by the Chien search, the following three conditions are used. (1) deg (Λ (Z)) ≦ t (2) L = deg (Λ (Z)) (3) deg (Λ (Z)) = (GF of Λ (Z)
(Number of zeros on (p ^m )) Decoding is possible if the above three conditions are all satisfied (the number of errors estimated to have occurred is t or less), and any one of the three conditions is not satisfied. Is undecodable (the number of errors estimated to have occurred is (t + 1) or more).

Similarly, when the error locator polynomial Λ (Z) and the error evaluator polynomial Ω (Z) are obtained by using the Euclidean method, the error locator polynomial Λ (Z) and the error evaluator polynomial Ω are obtained.
(Z) and the number of zeros of Λ (Z) obtained by the Chien search are used to perform the following three conditions. (1) deg (Λ (Z)) ≦ t (2) deg (Λ (Z))> deg (Ω (Z)) (3) deg (Λ (Z)) = (GF of Λ (Z)
(Number of zeros on (p ^m )) Decoding is possible if the above three conditions are all satisfied (the number of errors estimated to have occurred is t or less), and any one of the three conditions is not satisfied. Is undecodable (the number of errors estimated to have occurred is (t + 1) or more).

Further, in the algebraic decoding method called the transform decoding method in which the operation for obtaining the error polynomial of the algebraic decoding method (that is, step 4 and step 5) is changed, the decoding is executed in the next step. Step 1: Calculate 2t spectra (syndrome) from the receiving polynomial y (X) by inverse Fourier transform on a finite field. Step 2: If all syndromes are zero, it is determined that there is no error. Otherwise, go to step 3. Step 3: Obtain an error locator polynomial and an error evaluation polynomial from the syndrome. Step 4: Recursive relation between the coefficient of the error locator polynomial and the syndrome is used to recursively recombine the estimated error polynomial e (X) out of the n coefficients (spectra) of the Fourier transform E (Z) on the finite field. (N−2t) coefficients (spectrum) of are calculated. Step 5: An estimated error polynomial e (X) is calculated by performing an inverse Fourier transform on E (Z) on a finite field. Step 6: From the received polynomial y (X) to the estimated error polynomial e
(X) is subtracted and error correction is performed (the estimated code polynomial is calculated).

Steps 1, 2, and 3 are the same as the method of the usual algebraic decoding method. In step 4, the coefficient (spectrum) E _i of the Fourier transform E (Z) on the finite field of the estimated error polynomial e (X) is obtained. 2t of E _i are

E _i = e (α ⁱ ) = S _i (i = b, b + 1,
, B + 2t−1), the 2t syndromes (spectra) are used to calculate other (n−2t) coefficient E _i .

[Equation 27] (I = b + 2t, b + 2t + 1, ..., N-1, n (=
0), n + 1 (= 1), n + 2 (= 2), ..., N + b-
1 (= b-1)).

In step 5,

[Equation 28] By e _i = (1 / n) E (α ^-i ), that is, a polynomial E having E _i as a coefficient
The coefficient e _i of the estimation error polynomial e (X) is obtained by substituting α ⁱ (i = 0, 1, ..., N−1) into (Z) (that is, inverse Fourier transform on a finite field). Step 6
Is the same as that of the usual algebraic decoding method.

Since the transform decoding method does not require the Forney algorithm, there is no problem caused by the divider, which has been a problem in the usual algebraic decoding method.
However, even with this method, Λ in the ordinary algebraic decoding method
In addition to the coefficient calculation problem of (Z), there are problems in the following points. (1) in calculating all values of E _i, for the calculation of E _i is recursive, can not be calculated even higher E _i of order only after calculating the E _i of lower order, the calculation speed slow. (2) Since the error polynomial e (X) is calculated after obtaining all the coefficients of E (Z) from the syndrome, this is an algorithm that can be easily implemented by a circuit, but a fixed multiplication circuit of O (n ² ) is necessary. Of these two problems, (2)
Is the same problem as the circuit scale O (n ² ) of the Forney algorithm in the algebraic decoding method, and the biggest problem of the transform decoding method in comparison with the ordinary algebraic decoding method is (1). Limited.

The present invention has been made in view of such circumstances, and an object thereof is that the coefficient calculating circuits of the error locator polynomial Λ (Z) are the same and the delay time for obtaining the estimated error polynomial is constant (that is, calculation is performed). (EN) It is an object to provide a decoding method and a decoding circuit for a cyclic code (Reed-Solomon code, BCH code, etc.) in which the number of loop iterations is always a fixed number and condition determination is small. That is, it is to provide a decoding method which can be easily made into a circuit and a decoding circuit using the decoding method.

[0026]

A decoding circuit according to the present invention is a cyclic code decoding circuit capable of correcting t or less errors, and 2t calculated from a reception polynomial y (x) of the code string. Syndromes Si (i = b, b + 1, ..., b
+ 2t−1) based on the error locator polynomial coefficient Λi (i
= 1, 2, ..., ν; ν ≦ t), the error locator polynomial coefficient calculation circuit is provided, and the error locator polynomial coefficient calculation circuit is configured to perform decoding based on the coefficient Λi of the error locator polynomial. , The syndrome Si is arranged according to a predetermined rule in a t-row and t-column Hankel matrix (Hankel
matrix), an additional row in which the syndromes Si are arranged according to a predetermined rule is added, and a matrix of (t + 1) rows and t columns is applied with a matrix-reduction algorithm. To calculate.

The decoding circuit according to the present invention has t or less decoding circuits.
It is a cyclic code decoding circuit that can correct the error of
2t syndromes S calculated from the polynomial y (x)
error based on i (i = b, b + 1, ..., b + 2t-1)
Coefficient of the relocation polynomial Λi (i = 1, 2, ..., ν; ν ≦
error position polynomial coefficient calculation circuit for calculating t);
Based on the coefficient Si of the error Si and the error locator polynomial
Error evaluation polynomial for calculating the coefficient Ωi of the error evaluation polynomial
Equation coefficient calculation circuit and the coefficient Λi and the coefficient of the error locator polynomial
And the estimation error based on the coefficient Ωi of the error evaluation polynomial Ω (Z).
Fourier transform E (Z) on the finite field of the polynomial e (X)
A calculation circuit for calculating and the existence of the estimation error polynomial e (X)
Inverse Fourier transform of Fourier transform E (Z) on the field
A calculation circuit for calculating a constant error polynomial e (X),
Fourier transform on the finite field of the estimated error polynomial e (X)
The calculation circuit for calculating E (Z) is Λ (Z) Γ (Z) = 1
-Z ⁿCompute a complementary error locator polynomial Γ (Z) that satisfies
Error evaluation polynomial Ω (Z) and complementary error locator polynomial Γ (Z)
And by using the and on the finite field of the estimated error polynomial e (X)
The Rie transformation E (Z) is calculated.

At least one of the error locator polynomial coefficient calculating circuit and the error polynomial coefficient calculating circuit has the same coefficient calculating circuit and a constant delay time for obtaining the estimated error polynomial (that is, the calculation as described below). The number of loop repetitions is always a fixed number), the number of condition judgments is small, and the control circuit is unnecessary, so that the circuit configuration is further simplified.

The error locator polynomial coefficient calculation circuit is used for t rows and t
An additional row of 1 row and t columns is added to the Hankel matrix of columns (t
If the matrix-reduction algorithm is used for the matrix of +1) rows and t columns, the coefficient calculation circuit of the error locator polynomial is the same, and the number of iterations of the operation loop is always a certain number (the number of error correctable numbers: t times),
In addition, the number of condition judgments is reduced, and the increase in circuit size can be suppressed. The matrix-reduction algorithm itself is a known technique, and for example, ERBerlekam
p, “Algebraic Coding Theory”,
It is described in McGraw-Hill (1968) and the like.

Further, the coefficient E _i of the Fourier transform E (Z) on the finite field of the estimated error polynomial e (X) is not recursive, but the syndrome polynomial S (Z) and the error locator polynomial Λ.
By directly obtaining the error evaluation polynomial Ω (Z) calculated by (Z) and the polynomial multiplication of Γ (Z), E
The calculation of (Z) can be processed at high speed.

Let the coefficient e _i of the estimated error polynomial e (X) be E
By obtaining the estimated error polynomial e (X) directly by the inverse Fourier transform on the finite field of (Z), it is not necessary to allocate a divider according to the error position, so that a complicated control circuit is unnecessary. .

The present invention can be applied to any error correction circuit that employs a cyclic code (Reed-Solomon code, BCH code, etc.) in a digital communication system and a digital storage system.

[0033]

DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 shows a cyclic code according to the present invention (for example, Reed-Solomon code (RS code), BCH (Bose-Chaudhuri-Hocqenghem) code, etc.).
3 is a functional block diagram of the decoding circuit 10 of FIG. Reception polynomial y
The coefficients y _{0 to} y _n-1 of (X) are input to the syndrome calculation circuit 12 and the output circuit 14. Syndrome calculation circuit 12
Calculates the syndromes S _{b to} S _{b + 2t−1} based on the coefficients y _{0 to} y _n−1 of the reception polynomial y (X). The syndrome calculation circuit 12 is connected to the error position polynomial coefficient calculation circuit 16, and the error position polynomial coefficient calculation circuit 16 calculates the error position polynomial coefficients Λ _{1 to} Λν based on the syndromes S _{b to} S _{b + 2t-1.} To calculate. Note that ν (ν ≦ t) is the number of estimated errors. Further, the syndrome calculation circuit 12 and the error locator polynomial coefficient calculation circuit 16 are connected to the error evaluation polynomial coefficient calculation circuit 18, and the error evaluation polynomial coefficient calculation circuit 18 includes the syndromes S _{b to} S _b.
The coefficients Ω _{0 to} Ων ₋₁ of the error evaluation polynomial are calculated based on _{b + 2t−1} and the coefficients Λ _{1 to} Λν of the error locator polynomial. The error position polynomial coefficient calculation circuit 16 and the error evaluation polynomial coefficient calculation circuit 18 are connected to an error polynomial spectrum (Fourier transform on finite field) calculation circuit 20, and the error polynomial spectrum calculation circuit 20 Polynomial coefficients Λ _{1 to} Λν and error evaluation polynomial coefficients Ω _{0 to} Ων _-1
Then, the error polynomial spectra E _{0 to} E _n-1 are calculated based on Eq. The error polynomial spectrum calculation circuit 20 is connected to the error polynomial coefficient calculation circuit 22, and the error polynomial coefficient calculation circuit 22 is connected to the error polynomial spectrum E _{0 to} E _n-1.
The coefficients e ₀ to e _n-1 are calculated based on The output circuit 14 uses the coefficients e _{0 to} e _n−1 of the error polynomial e (X) input, and the reception polynomial y input via the delay circuit 24.
Code polynomial c in which errors of coefficients y _{0 to} y _n-1 of (X) are corrected
The coefficients c _{0 to} c _n-1 of (X) are output.

In the conventional decoding method, the circuit scale increases in accordance with the number of loop iterations, which increases in proportion to the error-correctable number t, and the number of condition judgments (branches). The error correctable number t is an error correction capability, and an increase in circuit scale due to this is inevitable. Therefore, in the present invention, the circuit scale is reduced by reducing the number of condition judgments (branches).

Further, in the conventional decoding method, even if the calculation circuits for the coefficients of the error locator polynomial are the same, the errors (and the number of errors) that are regarded as the occurrence of the number of iterations of the operation loop are generated.
However, since a plurality of condition judgments are made (or polynomial division is included), the circuit scale is increased. However, the present invention aims to provide a decoding circuit for a cyclic code in which the calculation circuit of the coefficient of the error locator polynomial is the same, the number of iterations of the operation loop is always a fixed number (error correctable number: t times), and the condition judgment is small. According to the present invention, the number of items to be controlled and conditions to be determined is reduced, so that the control circuit is simplified, which is advantageous over the conventional decoding method in terms of simplification of the circuit configuration and circuit scale. Further, the throughput of the decoding circuit of the present invention is such that the number of calculation loops is always a fixed number (the number of error corrections possible: t times).
Although the control is simplified as described above, the throughput of the decoding circuit according to the conventional decoding method is also determined by the error-correctable number t, and therefore the throughput of the decoding circuit of the present invention is the same as the throughput of the conventional decoding circuit.

The calculation contents to be executed for each element will be described below.

<Syndrome Calculation> The syndrome calculation is the same as the above-mentioned conventional technique.

<Re-Expression of Fourier Transform / Inverse Fourier Transform on Finite Field> First, the Fourier transform on a finite field is re-expressed by using a generating function for later description. In the concept of a generating function that includes a polynomial as one class, a generating function whose zero-order coefficient is nonzero always has an inverse element related to multiplication of a normal polynomial. Now, the elements a _i (i = 0,
1, 2, ..., N-1), the generating function (1-α ⁱ Z)
think of. Since a _i times the inverse element of this generating function is α ⁿ = 1, the coefficient as in the following Expression (13) becomes a polynomial of infinite degree which is periodic.

[Equation 29] This is a property of a generating function that does not require a condition for convergence in an infinite geometric series. When the generating function of the equation (13) is summed from 0 to n−1 for i, the following equation (14) is established.

[Equation 30] By multiplying both sides of this expression (14) by (1-Z ⁿ ), the following expression (15) is obtained.

[Equation 31] The frequency domain polynomial thus derived is obviously the Fourier transform A (Z) on the finite field of a (X). That is, the coefficient a _i and GF of the polynomial a (X) in the time domain
(P ^m) of the original, with undetermined original Z in the frequency domain by taking the sum of the generating function which is related to alpha ^i, deriving a Fourier transform A over a finite field of (X) (Z) You can

Similarly, the inverse Fourier transform on the finite field is re-expressed using the generating function. As in the case of Fourier transform on a finite field, attention is paid to the generating function of the following expression (16).

[Equation 32] When the generating function of this equation (16) is summed from 0 to n−1 for i, the following equation (17) is established.

[Expression 33] That is, the following equation (18) is obtained.

[Equation 34] Time-domain polynomial a (X) derived as described above
Is the inverse Fourier transform on the finite field of A (Z).

<Fourier Transform of Error Polynomial e (X) on Finite Field> Here, the Fourier transform of the error polynomial on finite field is re-expressed by a generating function. The error polynomial can be thought of as having zero error at the non-erroneous position. Therefore, there are ν non-zero error amounts e _h ,
Since the error amount is set to e _ji , the sum of ν of the length n is calculated to obtain the following expression (19).

[Equation 35] From this equation (19), the following equation (20) is obtained.

[Equation 36]

<Reexpression of Syndrome> Here, the expression (2
0) in the numerator part of the generating function

[Equation 37] Consider equation (21) multiplied by.

[Equation 38] This generating function includes a syndrome sequence in the low-order part of the frequency domain and takes a periodic infinite period sequence. Next, consider Expression (22) obtained by shifting the entire generating function by 2t.

[Formula 39] Here, these two generating functions (formula (21) and formula (2
By taking the difference of 2)), the following equation (23) is obtained. Formula (2
In 3), the coefficient parts of the 2t-th order and higher have the same value in each generating function, and cancel each other out, and only the 0-th to 2t−1-th order parts remain.

[Formula 40] What is important here is that the representation of the syndrome by the generating function is not the congruence relation by the remainder, but the Stronger by the equal sign.
It is an assertion (Stronger Assertion). The conventional syndrome expression was merely a Weaker Assertion by congruence. Here, the method of newly defining the syndrome by the generating function is the Stronger assertion which is equal in all orders from the 0th order coefficient to the infinite order coefficient. By this, a simultaneous equation that makes the coefficient zero can be determined.

(1) Calculation of error locator polynomial coefficient: When both sides of the operation formula (23) of the error locator polynomial coefficient calculation circuit 16 are multiplied by the error locator polynomial Λ (Z), the denominator of the generating function representation part on the left side. Since the part is one of the factors of the error locator polynomial, it is canceled by the error locator polynomial to obtain the following equation (24).

[Formula 41] Consider the relationship in the order of both sides of this equation (24). When the order is expressed in each term of Expression (24),

[Equation 42] Becomes The coefficient comparison on both sides is shown in FIG. FIG. 2 shows whether the coefficient of the corresponding order is non-zero. Therefore, it means that the equality holds when the terms from the νth order to the 2t−1th order are zero. That is, S (Z)
It needs to be zero in the range from ν order of Λ (Z) to 2t−1 order. However, assuming that the error number ν has not been derived, it is appropriate to assume the maximum error-correctable number t. Therefore, from t next to 2t-
Consider placing the coefficients of S (Z) Λ (Z) so that the coefficients up to the first order are zero. By doing so, Λ ₁ ,
For t unknowns of Λ ₂ , ..., Λ _{t, there} are t simultaneous equations. If the respective equations are independent of each other, it is guaranteed that the coefficients of the error locator polynomial can be determined. However, when ν errors occur, the number of linearly independent equations is ν, so it is necessary and sufficient that there are ν simultaneous equations for the unknown ν. However, here, the problem is solved by setting the maximum error-correctable number t to find a solution that allows Λ ν ₊₁ , ..., Λ _t of the unknowns Λ _i to be zero. In addition, this νth order equation must be an equation of the minimum order that can be represented by the obtained matrix. Here, from the t-th order of the equation (24), (2t-
1) The following equation (26) is established by comparing the following terms on both sides.

[Equation 43] Using the simultaneous equations of this equation (26) in the form of matrix product, S
It is represented by the product of the matrix of _i and Λ _i . At this time, when replacing the simultaneous equations with the form of the product of the matrix, by the arrangement of terms,
Several patterns are possible. Of which, Λ (Z)
The arrangement of the coefficients Λ _{i in ∑ i} makes a great difference in determining the final solution. In particular, the coefficients of Λ (Z) are arranged in ascending order (Λ ₁ , Λ ₂ , ..., Λν) and descending order (Λ
ν, Λ ν ₁ , ..., Λ ₁ ) are arranged in different ways, and the amount of time and effort required for final determination of the solution are different. In this specification, it is shown that the solution Λ (Z) of the minimum order can be derived by arranging in descending order. Therefore, the matrix expression of Expression (27) is used.

[Equation 44] This equation (27) can be simplified

[Equation 45] Will be written. However,

[Equation 46] And In particular, the matrix [S] is called the Hankel matrix.

Here, a calculation method of the coefficient Λ _i of the error locator polynomial Λ (Z) for simultaneously determining the number of errors and the error position will be given. First, the matrix of Expression (27) is converted into a form suitable for applying the matrix-reduction algorithm described later. Specifically, by transposing all of the right side of Expression (27) to the left side, the matrix [Λ] of 1 row and t column is newly added with −1 as (1 × (t + 1)) element and 1 row (t + 1 ) Column, and the matrix [S] having t rows and t columns is newly added to the (t + 1) th row as [−S _{b + t} −S _{b + t +1} ...
-Sb _{+ 2t-1} ] (= added row; augmented row) is added to form a matrix of (t + 1) rows and t columns, and the following Expression (28) is obtained.

[Equation 47] This formula (28) is simplified

[Equation 48] Can be expressed as This matrix

[Equation 49] The matrix [S] is the reduced triangular Aiden patent form (reduced traiangular
Repeat a series of operations consisting of the following four steps t times so as to obtain an idempotent form). This operation is a matrix
It is called the reduction algorithm.

<Matrix-Reduction Algorithm> Step 1: If the (1,1) element is non-zero, nothing is done. If the (1,1) element is zero, the column is replaced with another column. A column that can be replaced with the first column is a column in which the elements in the first row are non-zero and the diagonal elements are zero. If there is no such column, column replacement is performed with a column in which the element in the first row is nonzero and the diagonal element is nonzero. If all the elements in the first row are zero, do nothing. Step 2: Divide all elements in the first column by (1,1) element. If the (1,1) element is zero, do nothing. Step 3: A value obtained by multiplying the first column by the first row element of each target column is subtracted from the target column, and the elements in the first row other than the (1,1) element are set to zero. Step 4: Row rotation and column rotation.

Regarding the operation of the additional row [U] in the matrix-reduction algorithm described above, the processing performed by the basic column operation is similarly affected by the entire matrix, but only the square matrix [S] is affected by the row rotation. It becomes the operation target. queue

[Equation 50] On the other hand, when the above steps 1 to 4 are repeated t times, the part of the matrix [S] becomes a matrix called a reduced triangular aided potential form as shown in the following equation (29). However, S
_{Let i, j} represent the i-th row and j-th column element of the matrix.

[Equation 51] Here, δ _i (1 ≦ i ≦ in the diagonal element of Expression (29) is used.
t) is 0 or 1, and when it is 0, all the other elements in the column are 0, and when it is 1, all the other elements in the row are 0. For the obtained matrix of Expression (29), t rows and t
The matrix of equation (30) is obtained by subtracting the unit matrix of t rows and t columns from the portion of the column matrix [S].

[Equation 52] The matrix expressed by the linear combination of each row of Expression (30) and the sum of U ′ satisfies the simultaneous equations of Expression (28).

[Equation 53] Is the solution.

Here,

[Equation 54] , The row vector of the i-th row of [S] is shown, λ _i (Z) is

[Equation 55] It corresponds to the polynomial expressed as. Then, as a relation among these Λ (Z), λ _i (Z), and U ′ (Z), a general solution of an indefinite simultaneous equation has a degree of freedom of linear combination in each row, and thus a polynomial is used. In the expression, the following formula (3
It becomes like 2).

[Equation 56] Where k_iIs GF (p^m). This and
Equation (32) represents (p ^m)^t-all ν polynomials
Can be a candidate for Λ (Z). Also, as shown in equation (33)
Sea urchin_i(Z) is divisible by Λ (Z).

[Equation 57] However, the coefficient of true Λ (Z) has the smallest order in the linear combination of Expression (32). That is, λ _i
Equation (32) in which the coefficients k _{i of} are all zero becomes the minimum degree, and becomes the solution of Λ (Z).

[Equation 58]

<Conditions of Decodability / Impossible> Since the order of the error locator polynomial Λ (Z) obtained as described above is always t order or less, the conditions necessary for the judgment of decodability are: 1) deg (Λ (Z) )> deg (Ω (Z)) (2) deg (Λ (Z)) = GF of ^{(Λ (Z) (p m} )
The number of zeros above) ((2) is equivalent to "1-Z ⁿ is divided by Λ (Z)"). Decoding is possible when conditions (1) and (2) are both satisfied (the number of errors estimated to have occurred is t or less), and decoding is not possible if any one of conditions (1) and (2) is not satisfied. (The number of errors estimated to have occurred is (t + 1) or more). That is, the number of condition judgments is smaller than that of the conventional method (Peterson method, Beerkamp-Massie method, Euclidean method).

As described above, by defining the syndrome expression by the generating function, an indefinite simultaneous equation can be created from the order relation of the product of the syndrome polynomial and the error locator polynomial. The relationship by the matrix composed of forms the Hankel matrix.
From the algorithm for deriving the solution of the indeterminate simultaneous equations, the method that can determine the solution that is the true error locator polynomial is given. This method is only a method for obtaining the solution of simultaneous equations, and is a very easy-to-understand and simple error position deriving operation.

Note that step 1 of the matrix-reduction algorithm is realized by the circuit configuration shown in FIG. 3, for example, and step 3 is realized by the circuit shown in FIG. 5, for example.
The circuit shown in FIG. 3 is configured by combining an OR circuit (logical sum) 30, an AND circuit (logical product) 32, an inverter (negative) 34, and the like. FIG. 4 is a detailed view of a portion (square portion 36 in which S _{i, j} is written) omitted in FIG. 3 (an example in the case where S _{i, j} is an element of a finite field GF (2 ^m )). Further, in FIG. 5, reference numeral 40 denotes a multiplier on the finite field GF (2 ^m ), and 42 denotes an adder on the finite field GF (2 ^m ). The error-evaluation polynomial coefficient calculation circuit 18 uses the conventionally known method to generate the syndromes S _{b to} S _{b + 2t-1.}
And the coefficients Ω _{0 to} Ων ₋₁ of the error evaluation polynomial are calculated based on the coefficients Λ _{1 to} Λν of the error locator polynomial.

When the error position (error position polynomial) can be derived, the amount of error must be determined. For this, the error polynomial derivation derived by applying the idea of Fourier transform using a generating function By using the method, the error polynomial itself can be derived without deriving each error amount. The method will be described below. After briefly describing this method, the decoding circuit will be described.

(2) Calculation of error polynomial spectrum E _i : operation of error polynomial spectrum calculation circuit 20 In the present invention, the Fourier transform E (Z) itself of the estimated error polynomial e (X) on the finite field is obtained. , And the problem of the transform decoding method which is one of the algebraic decoding methods (1) (Because the calculation of E _i is recursive, the higher order E _i is not calculated _{until the} lower order E _i is calculated. The calculation speed is slow because it cannot be calculated.) The specific principle and method are shown below. Now, the following formula (3
Consider the polynomial S _n (Z) shown in 5).

[Equation 59] The 0th to 2t−1th order coefficients of S _n (Z) are the syndromes S _b , S _{b + 1} , ..., S _{b + 2t−1} obtained from the reception polynomial, and therefore can be known in advance at the time of decoding. It is a polynomial. Therefore, S _n (Z) in Expression (35) can also be expressed as Expression (36).

[Equation 60] At this time, the relationship between S _n (Z) and E (Z) is expressed by the following equation (3
It becomes like 7). However, [f (y)] _n is f
(Y) It represents mod (1-y ⁿ ).

[Equation 61] By using this S _n (Z), the following expression (38) is established.

[Equation 62] The following expression (39) is obtained by multiplying both sides of expression (38) by (1-Z ⁿ ).

[Equation 63] For an infinite periodic sequence that generates the sequence of S _n (Z) derived by the equation (39), Λ (Z)

[Equation 64] Considering the complementary error locator polynomial Γ (Z) having the following relationship, the equation (39) can be expressed by the following equation (41).

[Equation 65] Furthermore, by multiplying both sides of Expression (41) by Z ^b and taking the remainder by (1-Z ⁿ ), the following Expression (42) is obtained.

[Equation 66] On the left side of the equation (42), there is a product part of the error locator polynomial Λ (Z) shown in the equation (45) and the sum of the generating functions.

[Equation 67] this is,

[Equation 68] The error evaluation polynomial Ω (Z) of the equation (46) obtained by

[Equation 69] Therefore, the equation (43) is established.

[Equation 70] If the equation (43) can correctly derive the coefficient of the error locator polynomial Λ (Z), the estimation error using the error locator polynomial Λ (Z) and the error evaluator polynomial Ω (Z) without using the recursive relation of the syndrome. It is shown that the Fourier transform E (Z) of the polynomial e (X) can be derived and the estimation error polynomial e (X) can be directly derived by performing its inverse Fourier transform. That is, in calculating all the values of the transform decoding of the problems _{(1) (E i, E} i
Calculations for a recursive low the E _i of order since the E _i of further only after the calculated higher order can not be calculated, slow computational speed. ) Is E due to the fact of equation (43)
It is solved by immediately obtaining (Z).

Here, the calculation of the complementary error locator polynomial Γ (Z) is required, but the calculation of Γ (Z) requires 1-Z ⁿ to be Λ.
It is performed by dividing the polynomial by (Z). The remainder of this polynomial division is a zero polynomial (1-Z ⁿ
Is divided by Λ (Z)) and whether the error polynomial estimated to have occurred in the Peterson method, Pale-Kamp-Massie method, and Euclidean method is a decodable error polynomial. Condition (3) (deg
(Λ (Z)) = (the number of GF (p ^m ) zeros of Λ (Z))), and is obtained by the Peterson method, the Beerkamp-Massie method, and the Euclidean method.
The calculation is similar to the Chien search performed for (Z), and is an alternative calculation to the Chien search. However, the division of the polynomial here does not search the error position like the Chien search, so the error position cannot be calculated, but the Fourier transform on the finite field of the error polynomial is found by the formula (43), and the formula (43 There is no problem because the error polynomial e (X) itself is obtained by performing the inverse Fourier transform on E (Z) in 43) on the finite field. Further, in the polynomial division here, t fixed multiplications are repeated n times (t-th order polynomial division), but t fixed multiplications are repeated n times also in a commonly used Chien search. It is necessary, and since the calculation amount is similar, there is no problem from the viewpoint of calculation amount. As a result, one condition (1) (deg (Λ (Z))> is necessary for determining the remaining decodable / impossible.
deg (Ω (Z))) only.

(3) Calculation of error polynomial coefficient e _i : operation of error polynomial coefficient calculation circuit 22 As described above, the Fourier transform E (Z) of the estimated error polynomial e (X) on the finite field. Is calculated by the equation (43), the coefficient of the estimated error polynomial e (X) is obtained by calculating the inverse Fourier transform of this E (Z) on the finite field.

From the above results, it is also possible to successively derive the error amount at each error position, as used in the Forney algorithm. That is, the formula (4
As shown in 4), the error amount at the error position is derived by substituting the inverse element α ^-ji of the error position α ^ji into the undetermined element.

[Equation 71] With such sequential derivation, it is possible to derive the corresponding error amount without individually deriving the error position. That is, the process of calculating the error position and the process of calculating the error amount corresponding to the error position, which were conventionally performed by using the Chien search and the Forney algorithm, are simultaneously solved. Although the amount of calculation in this process is proportional to O (n ² ), the circuit is simplified because control such as designating an error position is unnecessary.

(4) Output: The operation of the output circuit 14 outputs the error-corrected code polynomial coefficients c _{0 to} c
Output _n-1 . As mentioned in the explanation of the prior art,

Since c (X) = y (X) −e (X), the coefficient e _i of the estimated error polynomial e (X) is subtracted from y _i to obtain the coefficient of the corrected code polynomial. c _i is obtained.

Although the preferred embodiment has been described above, it is merely an example, and various modifications are possible within an equivalent range. For example, it is possible to replace only the conventional error locator polynomial coefficient calculation circuit with the error locator polynomial coefficient calculation circuit (16) according to the present invention. In addition, a conventional error position and an error amount calculation circuit corresponding to the error position, and an error polynomial coefficient calculation circuit are provided as an error polynomial spectrum calculation circuit (20) and an error polynomial coefficient calculation circuit (22) according to the present invention. ) Is also possible.
However, in this case, it should be noted that the conventional error polynomial calculation circuit and the error polynomial calculation circuit (22) according to the present invention are different.

[0057]

As described above, the error locator polynomial Λ
A cyclic code decoding method and a decoding circuit in which the coefficient calculation circuit of (Z) is the same, the number of iterations of the operation loop is always a fixed number, and the condition determination is small are realized.

[Brief description of drawings]

FIG. 1 is a functional block diagram of a cyclic code decoding circuit according to the present invention.

FIG. 2 is a diagram for explaining coefficient comparison on both sides of Expression (24).

FIG. 3 is a circuit diagram (logic circuit) that realizes step 1 of the matrix-reduction algorithm.

FIG. 4 is a detailed view of a part omitted in FIG. 3 (a square part in which S _{i, j} is written) (an example when S _{i, j} is an element of a finite field GF (2 ^m )).

FIG. 5 is a circuit diagram (logic circuit) that implements step 3 of the matrix-reduction algorithm.

FIG. 6 is a functional block diagram of a conventional cyclic code decoding circuit.

[Explanation of symbols]

10 Decoding circuit, 12 Syndrome calculation circuit, 14
Output circuit, 16 Error position polynomial coefficient calculation circuit, 18
Error evaluation polynomial coefficient calculation circuit, 20 error polynomial spectrum calculation circuit, 22 error polynomial coefficient calculation circuit.

─────────────────────────────────────────────────── ───

[Procedure amendment]

[Submission date] December 12, 2001 (2001.12.
12)

[Procedure Amendment 1]

[Document name to be amended] Statement

[Correction target item name] 0009

[Correction method] Change

[Correction content]

Calculation of the syndrome in step 1 is performed by using GF
Original α of (p ^m) GF to the receiving polynomial y (X) on the (p ^{m)
It is performed} by substituting ^b , α ^{b + 1} , ..., α ^{b + 2t-1} (that is, Fourier transform on the finite field GF (p ^m )). That is,

[Equation 15] Calculated by For i such that b ≦ i ≦ b + 2t−1, always

Since c (α ⁱ ) = 0,

Since S _i = y (α ⁱ ) = c (α ⁱ ) + e (α ⁱ ) = e (α ⁱ ), there is no error if all 2t syndromes are 0 in step 2, If any of them is not 0, it is judged that there is an error.

Continued front page    (72) Inventor Yu Fujita             Maison Mona 3-5-1 Wakasato, Nagano City, Nagano Prefecture             Mi 206 (72) Inventor Takaki Shibata             5-1-1 Shimorenjaku, Mitaka City, Tokyo Japan             Wireless Co., Ltd. F-term (reference) 5B001 AA11 AB05 AC01 AE07                 5J065 AA01 AB01 AC01 AD11 AE06                       AF03 AG02 AH04

Claims (4)

[Claims] 1. A cyclic code decoding circuit capable of correcting t or less errors, wherein 2t syndromes Si (i = b, b + 1, ..., B +) calculated from a reception polynomial y (x).
2t−1) based on the error locator polynomial coefficient Λi (i =
1, 2, ..., ν; ν ≦ t), and a decoding circuit that performs decoding based on the coefficient Λi of the error locator polynomial, wherein the error locator polynomial coefficient calculation circuit is An additional row in which the syndromes Si are arranged according to a predetermined rule is added to a Hankel matrix of t rows and t columns in which the syndromes Si are arranged according to a predetermined rule (t +
1) Matrix-reduction (matr
ix-reduction) algorithm is applied to the matrix to calculate the error position polynomial coefficient Λi. 2. A cyclic code decoding circuit capable of correcting t or less errors, wherein 2t syndromes Si (i = b, b + 1, ..., b + 2t-1) calculated from a reception polynomial y (x). ), The coefficient Λi of the error locator polynomial Λ (Z) (i = 1, 2,
, Ν; ν ≦ t), and an error evaluation polynomial coefficient calculation circuit that calculates an error evaluation polynomial coefficient Ωi based on the syndrome Si and the coefficient Λi of the error position polynomial Λ (Z). A circuit and an estimated error polynomial e based on the coefficient Λi of the error locator polynomial Λ (Z) and the coefficient Ωi of the error evaluator polynomial Ω (Z).
A calculation circuit for calculating a Fourier transform E (Z) on the finite field of (X), and an estimated error polynomial e by inverse Fourier transforming the Fourier transform E (Z) on the finite field of the estimation error polynomial e (X). (X)
And a calculation circuit for calculating the Fourier transform E (Z) on the finite field of the estimation error polynomial e (X), Λ (Z) Γ (Z) = 1
Compute a complementary error locator polynomial Γ (Z) that satisfies −Z ⁿ ,
Error evaluation polynomial Ω (Z) and complementary error locator polynomial Γ (Z)
And a Fourier transform E (Z) on a finite field of the estimated error polynomial e (X) is calculated using. 3. A method of decoding a cyclic code capable of correcting t or less errors, wherein 2t syndromes Si (i = b, b +) calculated from a reception polynomial y (x) of the code string.
1, ..., B + 2t−1) based on the error position polynomial coefficient Λi (i = 1, 2, ..., ν; ν ≦ t), and decoding based on the error position polynomial coefficient Λi. In the decoding method of performing the step of calculating the coefficient of the error locator polynomial, the syndrome Si is arranged according to a predetermined rule in a Hankel matrix of t rows and t columns in which the syndrome Si is arranged according to a predetermined rule. Calculating a matrix of (t + 1) rows and t columns by adding the arranged additional rows, and applying a matrix-reduction algorithm to the matrix of (t + 1) rows and t columns, and matrix- Apply the reduction algorithm (t
+1) calculating the coefficient Λi of the error locator polynomial from the matrix of row t column, and a decoding method comprising: 4. A method of decoding a cyclic code capable of correcting t or less errors, wherein a reception polynomial y of a code string encoded by the cyclic code.
2t syndromes Si (i =
, b + 2t−1), the coefficient Λi (i = 1, 2, ..., ν; ν ≦ of the error locator polynomial Λ (Z) based on b, b + 1 ,.
t) and the coefficient Ωi of the error evaluator polynomial Ω (Z) based on the syndrome Si and the coefficient Λi of the error locator polynomial Λ (Z).
And the estimated error polynomial e based on the coefficient Λi of the error locator polynomial Λ (Z) and the coefficient Ωi of the error evaluator polynomial Ω (Z).
Calculating a Fourier transform E (Z) on the finite field of (X), and performing an inverse Fourier transform on the Fourier transform E (Z) on the finite field of the estimation error polynomial e (X) to estimate the error polynomial e ( X)
The step of calculating the Fourier transform E (Z) on the finite field of the estimated error polynomial e (X) comprises: Complementing Λ (Z) Γ (Z) = 1-Z ⁿ Calculating the error locator polynomial Γ (Z), the error evaluator polynomial Ω (Z) and the complementary error locator polynomial Γ (Z)
And a step of calculating a Fourier transform E (Z) on the finite field of the estimated error polynomial e (X) by using the following.