CN102394662A - Method for decoding broadcast channel (BCH) codes - Google Patents

Abstract

The invention discloses a method for decoding broadcast channel (BCH) codes. The method comprises the following steps of: S1, supplying BCH (30, 9, 12) codes, wherein the BCH (30, 9, 12) codes are acquired by adding a parity check bit onto shortened codes BCH (29, 9) of the original BCH (31,16); S2, constructing the generating polynomial of the BCH (30, 9, 12) codes, wherein the generating polynomial is as follows: g(X)=X20+X18+X17+X13+X10+X9+X7+X6+X4+X2+1; and in the generating polynomial, a, a2, a3,..., and a10 are used as roots; S3, obtaining a receiving polynomial r(X)=rn-1Xn-1+...r2X2+r1X1+r0 according to the generating polynomial in the step 2; S4, calculating a corrector S according to the receiving polynomial, wherein the corrector S=(S1,S2,..., and S2t); 5, determining an error polynomial sigma(X), wherein sigma(X)=(1+betavX)...(1+beta2X)(1+beta1X)=sigmavXv+...+sigma2X2+sigma1X+sigma0; and S6, calculating the root of the sigma(X), determining the number of error positions, and correcting the error of the r(X), wherein v is the number of errors; and beta1, beta2,..., and betav are the error positions.

Description

A kind of interpretation method of BCH code

Technical field

The present invention relates to the communications field of wireless communication system, specially refer to wireless with the interpretation method of broadcasting a kind of BCH code in the technology.

Background technology

In the DMR agreement, the 30ms data are a frame, comprise 288bit.With broadcasting in the technology, divide 6 time slots to send frame data at wireless digital, every time slot comprises the data volume of 5ms, i.e. 48bit.In the practical application, need have some data to be used for power and rise and distance protection, 18bit is used for this function in the present technique, and 30bit is an effective information.

One type of good linear error correction sign indicating number that BCH code is up to now to be found.Its error correcting capability is very strong, and under short and medium code length, its performance approaches theoretical value very much especially, and structure is convenient, and coding is simple.The decoding problem of BCH code is one of problem most interested in the coding theory research always.The inferior theoretical foundation of establishing the decoding of binary system BCH code of nineteen sixty Peter, dagger-axe thunder (Gore) and Ziller that (Zierler) is generalized to the multi-system situation after a while.Nineteen sixty-five good fortune Buddhist nun (Forney) has solved the error correction correcting and eleting codes of BCH code.Berlekamp had proposed iterative decoding algorithm in 1966, had saved amount of calculation, had accelerated decoding speed, thereby from fact having solved the decoding problem of BCH code.1969 Mei Xi (Massey) pointed out the line of shortest length property shift register of iterative decoding algorithm and the sequence relation between comprehensive, and simplify, henceforth call the BM iterative decoding algorithm to this decoding algorithm.BCH code decoding need be used the computing on the galois field.GF (2 ^m) in each unit have two kinds of method for expressing: power table shows and polynomial repressentation.Power table shows the power that is certain primitive element α, makes things convenient for multiplying.Polynomial repressentation is α ⁰, α ..., α ^M-1Linearity and form, make things convenient for add operation.

In the DMR agreement, the transmission of each time slot all needs 48 bit, and except 18 protection bits, 30 bit that also need be used for beared information to send in the lump, but under the actual conditions, have only 9 bit beared informations, can not satisfy the demand.

To this needs of DMR agreement, the inventor has designed BCH (30,9,12), through coding, changes to 30 bits to 9 bits, so just can send the desired data amount through a time slot.

Summary of the invention

In order to overcome the defective of prior art, the present invention proposes a kind of decoding algorithm of BCH code, its anti-interference is good, and can realize 30 bit beared informations.

Technical scheme of the present invention is following:

A kind of interpretation method of BCH code, it may further comprise the steps:

S1: a BCH (30,9,12) sign indicating number is provided, and said BCH (30,9,12) sign indicating number increases by a bit parity check position by the shortening sign indicating number BCH (29,9) of basis BCH (31,16) and gets;

S2: the generator polynomial of structure BCH (30,9,12) sign indicating number, said generator polynomial is:

G (X)=X ²⁰+ X ¹⁸+ X ¹⁷+ X ¹³+ X ¹⁰+ X ⁹+ X ⁷+ X ⁶+ X ⁴+ X ²+ 1, it is with α, α ², α ³..., α ¹⁰Be root;

S3: the generator polynomial according to S2 obtains receiverd polynomial r (X),

r(X)＝r _n-1X ^n-1+…r ₂X ²+r ₁X ¹+r ₀；

S4: by receiverd polynomial r (X) computing syndrome S, syndrome S=(S ₁, S ₂..., S _2t);

S5: confirm wrong multinomial σ (X) by syndrome S,

σ(X)＝(1+β _vX)…(1+β ₂X)(1+β ₁X)＝σ _vX ^v+…σ ₂X ²+σ ₁X+σ ₀；

S6: ask the root of σ (X), confirm the errors present number, and correct the mistake of r (X);

Wherein, v is an errors, β ₁, β ₂..., β _vBe errors present.

Preferably, said step S4 further comprises:

If send code word v (X)=v _N-1X ^N-1+ ... V ₂X ²+ v ₁X ¹+ v ₀, e (X) is an error pattern, then receiverd polynomial r (X) is r (X)=v (X)+e (X).Because of α, α ², α ³..., α ^2tFor generating root of polynomial, then syndrome does

S _i＝r(α ⁱ)＝r _n-1α ^(n-1)i+…r ₂α ²ⁱ+r ₁α ⁱ+r ₀

＝(…((r _n-1α ⁱ+r _n-2)α ⁱ+r _n-3)α ⁱ+…+r ₁)α ⁱ+r ₀ 1≤i≤2t

These computational methods are optimum under software is realized; Polynomial f (X) on the GF (2) has f ²(X)=f (X ²), then syndrome satisfies S _2i=S _i ²

Preferably, said step S5 further comprises:

σ _iAnd the relation between the syndrome S is confirmed by following Newton's identities:

S ₁+σ ₁＝0

S ₂+σ ₁S ₁+2σ ₂＝0

.

.

.

S _v+σ ₁S _v-1+…+σ _v-1S ₁+vσ _v＝0

S _v+1+σ ₁S _v+…+σ _v-1S ₂+σ _vS ₁＝0

Above equation group has much separates, and desired is separating of σ (X) with minimum number of times.

Preferably, said step S5 uses the BM iterative algorithm of simplifying to find the solution σ (X):

Make μ=-1/2,0,1,2 ... T is an iteration step, σ ^(μ)(X) be the wrong multinomial of the μ time iteration acquisition, d _μBe μ step difference, l _μBe σ ^(μ)(X) number of times.When μ=-1/2 went on foot, initial value was σ ^(1/2)(X)=1, d _μ=1.When μ=0 went on foot, initial value was σ ⁽⁰⁾(X)=1, d _μ=S ₁Suppose and accomplished μ step iteration, μ+1 step is:

1) if d _μ=0, σ then ^(μ+1)(X)=σ ^(μ)(X)

2) if d _μ≠ 0, obtain the step ρ of μ before the step and make d _ρ≠ 0 and 2 ρ-l _ρHas maximum.So

σ ^(μ+1)(X)＝σ ^(μ)(X)+d _μd _ρ ^-1X ^2(μ-ρ)σ ^(ρ)(X)

Which kind of situation no matter, difference d _μ+1For

$d_{\mathrm{\&mu;}+1}={\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+3}+{{\mathrm{\&sigma;}}_1}^{(\mathrm{\&mu;}+1)}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+2}+{{\mathrm{\&sigma;}}_2}^{(\mathrm{\&mu;}+1)}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+1}+\&CenterDot;\&CenterDot;\&CenterDot;+{{\mathrm{\&sigma;}}_{l_{\mathrm{\&mu;}+1}}}^{(\mathrm{\&mu;}+1)}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+3-l_{\mathrm{\&mu;}+1}}$

Multinomial σ in last column ^(t)(X) be exactly the σ (X) that requires; If its number of times, then has the mistake that can not entangle more than t greater than t in receiverd polynomial r (X).

Preferably, as errors v during, need not carry out t step iteration and just can obtain σ (X) less than t;

If for certain μ, d _μAnd thereafter

The step difference of iteration all is 0, then σ ^(μ)(X) be exactly error location polynomial; If error number v≤t then only needs

The step iteration.

Preferably, said step S6 further comprises:

With 1, α, α ²..., α ^N-1(n=2 ^m-1) substitution σ (X) just can try to achieve the root of σ (X).Because α ⁿ=1, so α ^-l=α ^N-l, if α ^lBe the root of σ (X), then α ^N-lBe exactly the errors present number, and receive position r _N-lIt is error bit.

Preferably, among the said step S6, m=5, t=5; As errors v during less than t, the step iteration can obtain wrong multinomial σ (X) only to need

.

Compared with prior art, anti-interference of the present invention is good, and can realize 30 bit beared informations.

According to wireless digital with data format and the strong error-correcting performance of BCH code broadcast in the technology, so designed BCH (30,9,12) yard.Because the transmission of each time slot all needs 48 bit, and except 18 protection bits, also need 30 bit to come to send in the lump, but under the actual conditions, have only 9 bit beared informations, behind BCH (30,9,12) coding, these 9 bit have just become 30 bit.In sum, the purpose of structure BCH (30,9,12) sign indicating number has two: the first, changes to 30bit to 9bit, the bit number of symbol emission; The second, the performance of BCH code self is fine, and is anti-interference strong.

Description of drawings

Soft, the hard decoding mistake of Figure 1B CH (30,9,12) is believed relatively sketch map of performance;

Fig. 2 BCH (30,9,12) is soft, decipher relatively sketch map of flase drop performance firmly.

Embodiment

The below combines accompanying drawing and specific embodiment that the present invention is done further description.

Embodiment

A kind of interpretation method of BCH code, it may further comprise the steps:

S1: a BCH (30,9,12) is provided sign indicating number, and BCH (30,9,12) sign indicating number belongs to GF (2 ⁵) territory, said BCH (30,9,12) sign indicating number increases by a bit parity check position by the shortening sign indicating number BCH (29,9) of basis BCH (31,16) and get, BCH (30,9,12) can correct smaller or equal to 5 wrong.

S2: the generator polynomial of structure BCH (30,9,12) sign indicating number, said generator polynomial is:

G (X)=X ²⁰+ X ¹⁸+ X ¹⁷+ X ¹³+ X ¹⁰+ X ⁹+ X ⁷+ X ⁶+ X ⁴+ X ²+ 1, it is with α, α ², α ³..., α ¹⁰Be root.

S3: the generator polynomial according to S2 obtains receiverd polynomial r (X),

r(X)＝r _n-1X ^n-1+…r ₂X ²+r ₁X ¹+r ₀；

S4: by receiverd polynomial r (X) computing syndrome S, syndrome S=(S ₁, S ₂..., S _2t);

S5: confirm wrong multinomial σ (X) by syndrome S,

σ(X)＝(1+β _vX)…(1+β ₂X)(1+β ₁X)＝σ _vX ^v+…σ ₂X ²+σ ₁X+σ ₀；

S6: ask the root of σ (X), confirm the errors present number, and correct the mistake of r (X);

Wherein, v is an errors, β ₁, β ₂..., β _vBe errors present.

In the present embodiment, said step S4 further comprises:

If send code word v (X)=v _N-1X ^N-1+ ... V ₂X ²+ v ₁X ¹+ v ₀, e (X) is an error pattern, then receiverd polynomial r (X) is r (X)=v (X)+e (X).Because of α, α ², α ³..., α ^2tFor generating root of polynomial, then syndrome does

S _i＝r(α ⁱ)＝r _n-1α ^(n-1)i+…r ₂α ²ⁱ+r ₁α ⁱ+r ₀

＝(…((r _n-1α ⁱ+r _n-2)α ⁱ+r _n-3)α ⁱ+…+r ₁)α ⁱ+r ₀?1≤i≤2t

These computational methods are optimum under software is realized; Polynomial f (X) on the GF (2) has f ²(X)=f (X ²), then syndrome satisfies S _2i=S _i ²

In the present embodiment, said step S5 further comprises:

Step S5 confirms wrong multinomial σ (X) by syndrome S: derive through theoretical, can get

σ _iAnd the relation between the syndrome S is confirmed by following Newton's identities:

S ₁+σ ₁＝0

S ₂+σ ₁S ₁+2σ ₂＝0

.

.

.

S _v+σ ₁S _v-1+…+σ _v-1S ₁+vσ _v＝0

S _v+1+σ ₁S _v+…+σ _v-1S ₂+σ _vS ₁＝0

Above equation group has much separates, and desired is separating of σ (X) with minimum number of times.Use the BM iterative algorithm of simplifying to find the solution σ (X) below.

Make μ=-1/2,0,1,2 ... T is an iteration step, σ ^(μ)(X) be the wrong multinomial of the μ time iteration acquisition, d _μBe μ step difference, l _μBe σ ^(μ)(X) number of times.When μ=-1/2 went on foot, initial value was σ ^(1/2)(X)=1, d _μ=1.When μ=0 went on foot, initial value was σ ⁽⁰⁾(X)=1, d _μ=S ₁Suppose and accomplished μ step iteration, μ+1 step is:

1) if d _μ=0, σ then ^(μ+1)(X)=σ ^(μ)(X)

2) if d _μ≠ 0, obtain the step ρ of μ before the step and make d _ρ≠ 0 and 2 ρ-l _ρHas maximum.So

σ ^(μ+1)(X)＝σ ^(μ)(X)+d _μd _ρ ^-1X ^2(μ-ρ)σ ^(ρ)(X)

Which kind of situation no matter, difference d _μ+1For

$d_{\mathrm{\&mu;}+1}={\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+3}+{{\mathrm{\&sigma;}}_1}^{(\mathrm{\&mu;}+1)}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+2}+{{\mathrm{\&sigma;}}_2}^{(\mathrm{\&mu;}+1)}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+1}+\&CenterDot;\&CenterDot;\&CenterDot;+{{\mathrm{\&sigma;}}_{l_{\mathrm{\&mu;}+1}}}^{(\mathrm{\&mu;}+1)}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000600787200011.png"\; state="\{\{state\}\}">S</figure-callout>}}_{2\mathrm{\&mu;}+3-l_{\mathrm{\&mu;}+1}}$

Multinomial σ in last column ^(t)(X) be exactly the σ (X) that requires; If its number of times, then has the mistake that can not entangle more than t greater than t in receiverd polynomial r (X).

The iterations of this simplification BM algorithm is t, is the half the of common BM algorithm, but is only applicable to binary BCH code.As errors v during, need not carry out t step iteration and just can obtain σ (X) less than t.If for certain μ, d _μAnd thereafter

The step difference of iteration all is 0, then σ ^(μ)(X) be exactly error location polynomial; If error number v≤t then only needs

The step iteration.

In the present embodiment, said step S6 is the calculating of errors present number: the errors present number is the inverse of σ (X) root.Said step S6 further comprises:

With 1, α, α ²..., α ^N-1(n=2 ^m-1) substitution σ (X) just can try to achieve the root of σ (X).Because α ⁿ=1, so α ^-l=α ^N-l, if α ^lBe the root of σ (X), then α ^N-lBe exactly the errors present number, and receive position r _N-lIt is error bit.

BCH (30,9,12) sign indicating number is used the BM iterated algorithm decoding of above-mentioned simplification, m=5 here, t=5; As errors v during less than t, the step iteration can obtain wrong multinomial σ (X) only to need

.

In the decode procedure, parity check bit is not participated in error correction, thus might cause when carrying out parity check a large amount of correct sign indicating numbers by misdirection, signal to noise ratio preferably time the especially, thus cause mistake to believe the reduction of performance.The soft Reliability of Information alpha of parity check bit capable of using determines whether carrying out parity check, and given thresholding door when alpha＞door, carries out parity check.The soft information of parity check bit is carried out the 8bit level quantize, be designated as parity, its hard decision is hard, and then confidence level alpha does

alpha＝abs(parity-128)+hard-1

Obtaining preferable threshold value through emulation is door=91.

In the 4fsk modulation system, under the white Gaussian noise channel BCH (30,9,12) decoding performance is carried out emulation, data volume is 10 ⁵Frame.Fig. 1 believes performance relatively for soft, hard decoding mistake, Fig. 2 for soft, decipher the flase drop performance relatively firmly, wherein hard decision comprises and uses the soft information credibility of parity check bit not regulate with not using the soft information credibility of parity check bit to regulate.

Among Fig. 1, abscissa is a signal to noise ratio, i.e. the ratio of information and noise, reflection be the power of ambient signal.Ordinate is represented mistake letter rate, and mistake letter rate just is meant because the influence of channel noise causes signal in the process of transmission, can make a mistake.

Among Fig. 2, abscissa is a signal to noise ratio, i.e. the ratio of information and noise, reflection be the power of ambient signal.Ordinate is represented false drop rate.Finding that when receiving verification wrong probability is exactly to miss the letter rate, when verification, do not find mistake, is exactly false drop rate with real data probability devious still.

Under identical signal to noise ratio, mistake letter rate and false drop rate are more little good more.

When guaranteeing decoding performance, need to consider operand.Table 1 has provided operation clock cycle (clock) and the operation time (ms) of BCH (30,9,12) coding and decoding under the CCS environment, and frequency is 48MHz.

Table 1

	Coding	Hard decoder	Soft decoding
				Clock cycle (clock)	872	35548	286084
Operation time (ms)	18(μs)	740(μs)	6(ms)

The preferred embodiment of the present invention just is used for helping to set forth the present invention.Preferred embodiment does not have all details of detailed descriptionthe, does not limit this invention yet and is merely described embodiment.Obviously, according to the content of this specification, can do a lot of modifications and variation.These embodiment are chosen and specifically described to this specification, is in order to explain principle of the present invention and practical application better, thereby person skilled can be utilized the present invention well under making.The present invention only receives the restriction of claims and four corner and equivalent.

Claims (7)

1. the interpretation method of a BCH code is characterized in that, it may further comprise the steps:

S1: a BCH (30,9,12) sign indicating number is provided, and said BCH (30,9,12) sign indicating number increases by a bit parity check position by the shortening sign indicating number BCH (29,9) of basis BCH (31,16) and gets;

S2: the generator polynomial of structure BCH (30,9,12) sign indicating number, said generator polynomial is:

G (X)=X ²⁰+ X ¹⁸+ X ¹⁷+ X ¹³+ X ¹⁰+ X ⁹+ X ⁷+ X ⁶+ X ⁴+ X ²+ 1, it is with α, α ², α ³..., α ¹⁰Be root;

S3: the generator polynomial according to S2 obtains receiverd polynomial r (X),

r(X)＝r _n-1X ^n-1+…r ₂X ²+r ₁X ¹+r ₀；

S4: by receiverd polynomial r (X) computing syndrome S, syndrome S=(S ₁, S ₂..., S _2t);

S5: confirm wrong multinomial σ (X) by syndrome S,

σ(X)＝(1+β _vX)…(1+β ₂X)(1+β ₁X)＝σ _vX ^v+…σ ₂X ²+σ ₁X+σ ₀；

S6: ask the root of σ (X), confirm the errors present number, and correct the mistake of r (X);

Wherein, v is an errors, β ₁, β ₂..., β _vBe errors present.

2. the interpretation method of a kind of BCH code according to claim 1 is characterized in that, said step S4 further comprises:

If send code word v (X)=v _N-1X ^N-1+ ... V ₂X ²+ v ₁X ¹+ v ₀, e (X) is an error pattern, then receiverd polynomial r (X) is r (X)=v (X)+e (X).Because of α, α ², α ³..., α ^2tFor generating root of polynomial, then syndrome does

S _i＝r(α ⁱ)＝r _n-1α ^(n-1)i+…r ₂α ²ⁱ+r ₁α ⁱ+r ₀

＝(…((r _n-1α ⁱ+r _n-2)α ⁱ+r _n-3)α ⁱ+…+r ₁)α ⁱ+r ₀ 1≤i≤2t

These computational methods are optimum under software is realized; Polynomial f (X) on the GF (2) has f ²(X)=f (X ²), then syndrome satisfies S _2t=S _i ²

3. the interpretation method of a kind of BCH code according to claim 1 is characterized in that, said step S5 further comprises:

σ _iAnd the relation between the syndrome S is confirmed by following Newton's identities:

S ₁+σ ₁＝0

S ₂+σ ₁S ₁+2σ ₂＝0

.

.

.

S _v+σ ₁S _v-1+…+σ _v-1S ₁+vσ _v＝0

S _v+1+σ ₁S _v+…+σ _v-1S ₂+σ _vS ₁＝0

Above equation group has much separates, and desired is separating of σ (X) with minimum number of times.

4. the interpretation method of a kind of BCH code according to claim 3 is characterized in that, said step S5 uses the BM iterative algorithm of simplifying to find the solution σ (X):

Make μ=-1/2,0,1,2 ... T is an iteration step, σ ^(μ)(X) be the wrong multinomial of the μ time iteration acquisition, d _μBe μ step difference, l _μBe σ ^(μ)(X) number of times.When μ=-1/2 went on foot, initial value was σ ^(1/2)(X)=1, d _μ=1.When μ=0 went on foot, initial value was σ ⁽⁰⁾(X)=1, d _μ=S ₁Suppose and accomplished μ step iteration, μ+1 step is:

1) if d _μ=0, σ then ^(μ+1)(X)=σ ^(μ)(X)

2) if d _μ≠ 0, obtain the step ρ of μ before the step and make d _ρ≠ 0 and 2 ρ-l _ρHas maximum.So

σ ^(μ+1)(X)＝σ ^(μ)(X)+d _μd _ρ ^-1X ^2(μ-ρ)σ ^(ρ)(X)

Which kind of situation no matter, difference d _μ+1For

$d_{\mathrm{\&mu;}+1}=S_{2\mathrm{\&mu;}+3}+{{\mathrm{\&sigma;}}_1}^{(\mathrm{\&mu;}+1)}{S}_{2\mathrm{\&mu;}+2}+{{\mathrm{\&sigma;}}_2}^{(\mathrm{\&mu;}+1)}{S}_{2\mathrm{\&mu;}+1}+\&CenterDot;\&CenterDot;\&CenterDot;+{{\mathrm{\&sigma;}}_{l_{\mathrm{\&mu;}+1}}}^{(\mathrm{\&mu;}+1)}{S}_{2\mathrm{\&mu;}+3-l_{\mathrm{\&mu;}+1}}$

Multinomial σ in last column ^(t)(X) be exactly the σ (X) that requires; If its number of times, then has the mistake that can not entangle more than t greater than t in receiverd polynomial r (X).

5. the interpretation method of a kind of BCH code according to claim 4 is characterized in that,

As errors v during, need not carry out t step iteration and just can obtain σ (X) less than t;

If for certain μ, d _μAnd thereafter

The step difference of iteration all is 0, then σ ^(μ)(X) be exactly error location polynomial; If error number v≤t then only needs

The step iteration.

6. the interpretation method of a kind of BCH code according to claim 1 is characterized in that, said step S6 further comprises:

With 1, α, α ²..., α ^N-1(n=2 ^m-1) substitution σ (X) just can try to achieve the root of σ (X).Because α ⁿ=1, so α ^-1=α ^N-l, if α ^lBe the root of σ (X), then α ^N-lBe exactly the errors present number, and receive position r _N-lIt is error bit.

7. according to the interpretation method of claim 5 or 6 described a kind of BCH codes, it is characterized in that, among the said step S6, m=5, t=5; As errors v during less than t, the step iteration can obtain wrong multinomial σ (X) only to need .

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