JPS6165534A - Error correction coding and decoding system - Google Patents

Abstract

PURPOSE:To correct a burst error having a large rate of correctable burst length to a code length and less weight by using a feedback shift register constituting a specific generation polynomial to attain coding/decoding for error correction. CONSTITUTION:A generation polynomial constituting a feedback shift register is obtained as a product between the 1st generation polynomial comprising cyclic codes having >=2 information symbols having the capability of correcting all random errors below (t) (where t is an optional integer >=3) and the 2nd generation polynomial comprising cyclic codes whose period is different from the measure of the period of the 1st generation polynomial having a minimum distance of (t+1) or over and >=2 information symbols. When an information series is inputted from an input terminal 201, the division is executed by a feedback shift register, a check bit is generated. Further, the received series is inputted to buffer registers 521-625 and also inputted to the feedback shift register, where division is conducted. At the end of input of the received series, the syndrome is calculated and the error of the received information series is corrected by using it.

Description

[Detailed Description of the Invention] [Field of Application of the Invention] The present invention relates to an error correction encoding and decoding system, and in particular to a practical low-weight burst error correction encoding and decoding system with a high degree of freedom in parameter selection. be.

[Background of the invention]

Recently, in the transmission and recording of digital signals, improving 48 reliability has become one of the most important issues, and various applications of error correction codes have been proposed.

Errors that actually occur in digital transmission and recording are:
Mainly burst errors with small weights.

However, with conventional encoding/decoding methods, even when creating a code that corrects only errors with a small weight (number of erroneous symbols in one burst), the degree of freedom in selecting code parameters is small; A practically effective code could not be obtained. Therefore, conventionally, codes that correct all burst errors or random error correction codes have been used.

On the other hand, efficient encoding/decoding methods for correcting burst errors with small weights have been proposed in the past (for example, IEEE Trans.

Informat, ionTreeory+
I T 9, P, 124 (1963) A, D, Wyner Low Densi Burst Correcting Code
5J). The outline described in the above document is as follows.

Now, assume that p(x) is an irreducible polynomial of degree r over the Galois field GF(q), and the order of its root is nl. Also, let 1g1(x) be a generator polynomial of a cyclic code with code length n21 j and short distance 2t+1.

At this time, a polynomial obtained by multiplying p(x) and gl(x) generates a code string different from either of these.

f (x) = g t (x) ・p (x)
-(1) The code generated by the above equation (1) is r≧b.

If n2≧2b-1, the weight is less than or equal to the burst length.

It is possible to correct burst errors when the sum is less than or equal to t.

The code length n of this code is given by the least common multiple of nl and nl, but in order to increase the ratio of the number of information symbols to the code length n, that is, the efficiency, it is necessary to make nl and nl relatively prime. There is. In this case, since the code for correcting burst errors of length b or less has a code length of nr(2b-1), the ratio of the correctable burst length to the code length is small. This is because the code length n of the code generated by gl(x)
2 and the order r of p(x), n 2 = 2
r-1 is established.

r=b holds true. Furthermore, since the code length is nl Xn2, nr·n2=nl(2b-1) holds true.

For example, if the correctable burst length is 10, then nl
>1000, the code lengths n1 and nl are several thousand bits in length. Even with such a long code string,
This means that only a burst length of 10 bits can be corrected.

[Purpose of the invention]

An object of the present invention is to improve such conventional drawbacks, and to provide a system in which the ratio of the correctable burst length to the code length is relatively large in transmission lines and recording devices where burst errors occur.
Another object of the present invention is to provide an error correction encoding and decoding system that can correct burst errors with small weights.

[Summary of the invention]

In order to achieve the above object, the error correction encoding and decoding method of the present invention has the ability to correct all random errors of (t is an arbitrary integer constant of 3 or more) or less information symbols with a number of 2 or more. a first generator polynomial of the cyclic code, and a second generator polynomial of the cyclic code, which has a minimum distance (t + 1) or more and the number of information symbols is 2 or more, and whose period is different from a divisor of the period of the first generator polynomial. A feedback shift register is constructed by using the product of - It is characterized in that it is input to a shift register, a syndrome is calculated, and the syndrome is used to correct errors in the received information sequence.

[Embodiments of the invention]

Before giving examples, the present invention will be explained in detail.

In the following explanation, in order to make it easier to understand, the codes on GF(2) will be described, but it goes without saying that the codes can generally be extended to codes on GF(q).

Now, let gt(X) be the order m1. It is assumed that the generator polynomial is a code having a code length n1 and a minimum distance of 2t+1 or more. Also,
g2(x) with order m2. Minimum distance t+1 with code length n2
Assume that the above code is a generating polynomial. At this time, consider the generator polynomial generated by multiplying both.

g (x) = g l (x) ・g 2 (x)
-(2) Code length n generated by the above formula (2)
The code C of = LCM(nlln2) is the burst length b=min.

('n' J+n:z) or less and that all burst errors with a weight of t or less can be corrected will be explained next.

In addition, L-CM(xl + x2) is xl and x2
The least common multiple of n+in, (xt l X2 ) is
rxJ represents the minimum value of χ1 and x2, and the maximum integer not exceeding χ.

Now, let xiA(x) and jB(x) be arbitrary burst errors whose weight is less than or equal to the burst length and whose weight is less than or equal to t.

In A (x) and B(x), the coefficient of the zero-order term is 1, and the order of the highest-order term does not exceed b−t.

In order to prove the error correction ability of the code C, it will be shown that the sum of the burst errors is not divisible by 1 when divided by the generator polynomial. That is, U<x)=xiA(
x) + xjB(x) - (3) If the above equation (3) is x” A (x)≠xjB(x), then g(X)
By showing that it is not divisible by , we prove that it has error correction ability.

Now, it is assumed that the linear equation holds true without losing the film property.

j≧1 + 3-1 = 'T n 1+α, 0≦2〈
nl...(4) Substituting the above equation (4) into the final equation of equation (3) yields the following.

x B(x)=x”””+18(x)i+l
i+j =X B(X)+X B(X)+Xqn”1B
(X) = X"' B(X) + X"'(X"1
+1) (3(x) Therefore, equation (3) can be expressed as follows.

U (x) = Xl(A(X)+x'f3(x)
Dx”(xqn' +1)B(x)...(5) The code length, that is, the period, of the code generated by gz(x) is n
Since l, gt (xH knee) J xl+l (x
qn1+1) is divisible. Therefore, if the first term on the right side of the above equation (5) is not divisible, it has a correction ability. Let's divide it into three parts depending on the value of Q.

■If ≦α≦“-21-”-1, x'B(x) has only terms whose degree is greater than or equal to Q and less than or equal to Q+b-1, and A(x
) has a term of order O, and A(x)+x'B
Since the weight of (x) is less than or equal to 2, xi(A(x
)+x'B(x)) means that (x) has a term of order 2 and A(x) does not have a term of order Q due to gt(X),
and (A(x)=x'B(x)) nod(X
Since ``1+1) is less than or equal to weight 2, xl(A(
x)+x'B(x)) is not divisible by gl(x). In addition, in the case of n=O, if A(x)≠B(x), the weight of A(x)+B(x) is greater than or equal to 1 and less than or equal to 2, so xJA(x)+B(x)) is not divisible by gt(x).

Therefore, U (x) is n=O and A (x
)=B(x), it is divisible by Φgl(x).

If α=0 and A(x)=B(x), then g (x)
The period of is n = L-C-M (n1 + n2),
Since and gl(x) is a generator polynomial of a code with a code length n2 and a minimum distance t+1 or more, U (x ) =
X1+1(X(Ln+ +l)B (x) is q≠0
Then, it is not divisible by gl(x).

From the above, both equations (3) are xiA(x)≠xjB
(x), it is not divisible by g(x).

This proves the error correction ability of code C.

The present invention proposes a method of creating a code using the generator polynomial given by both equations (2) and a method of decoding it.
It is possible to have the ability to correct all burst errors with a weight of less than or equal to b and less than or equal to weight t.

Next, an example of code C generated according to the present invention will be shown.

Let gl(x) be a generator polynomial of a triple random error correction BCH code with code length n 1 = 15, and be expressed by the following equation.

gl(x) = (x4+x+ t>(x' +x3+x
2+x+ 1)・(x2+x+ 1)=x” +x8
+x"+x' +x2+x+ 1
-(6) Also, let gs+(x) be a generator polynomial of a code with code length n2=7 and minimum distance 4, and be expressed by the following equation.

gl(x)=(x3+x+ 1)(xt 1)=x'
+x3+x2+ 1 .=(7) The generator polynomial generated by multiplying the generator polynomials in equations (6) and (7) above is as follows.

g(x)−gt(x)・g2(x)=x”
+x 13 +x ” +x”' +x8
+xS +x+1...(8
) The code C1 generated by the above equation (8) has a code length of 10
5. The number of check symbols is 14, and all burst errors with a burst length of 7 or less and a weight of 3 or less can be corrected.

Embodiments of the present invention will be described below with reference to the drawings.

FIG. 1 is a block diagram of an error correction encoding circuit showing an embodiment of the present invention.

In FIG. 1, 101 to 114 represent information symbols using the above formula (
8) is stored in the register where the exponent of each term of the remainder corresponds to the last 1 to 2 digits of the register number. 3
01 to 307 are exclusive OR circuits, and 401 and 402 are changeover switches.

An information sequence is input from the input terminal 201 from time O to time 90. At this time, the switch 401 is on, and division by 1 is executed by the feedback shift register configured by the generator polynomial in equation (8) above. That is, the input information series is transmitted to the circuit 30
7, the exclusive OR with X14 of the register 114 is taken, and through the switch 401, each exclusive OR circuit 30
Input from 1 to 306. When the input signal and X in register 101 are exclusive ORed, this result is input to register 102, and the input signal and x in register 105 are input to register 102.
When the exclusive OR with 5 is taken, this result is input to the register 106, and the other exclusive ORs are taken in the same way.

By repeating this 90 times, check bits are generated.

Furthermore, since the changeover switch 402 is connected to the terminal 204 side, the information series is output from the output terminal 202. From time 91 to time 104, switch 401
is off, and the selector switch 402 is switched to the terminal 203 side, so that the inspection symbol generated by the feedback shift register is output from the output terminal 202.

FIG. 2 is a block diagram of an error correction decoding circuit showing an embodiment of the present invention.

The registers 501 to 514 store the remainder obtained by dividing the received sequence by the generator polynomial of both equations (8), and the exponent and the last 1 to 2 digits of the register number correspond.

The received sequence is input to the buffer registers 521-625 and at the same time is input to the feedback shift register where division is performed.

At the end of inputting the received sequence, the syndrome is calculated and stored in the feedback shift register.

The pattern discriminator 800 has a burst length of 7 or less and a weight of 3.
It is necessary to identify the following single burst errors from syndromes. The pattern discriminator 800 is designed to generate a "1" on the next shift each time it recognizes the contents of the feedback shift register that match the error pattern in the last register position @625 of the received series. ing.

Next, the code cl used in FIGS. 1 and 2.

This is comparable to conventional codes that have the same level of error correction ability.

Conventionally, this code corrects all burst errors with a burst length of 7 or less and a weight of 3 or less.

A triple random error correction code and a code that corrects all burst errors with a burst length of 7 or less have been considered.1 As a representative example of both, a code length of 127° and a code with a number of check symbols of 21 are considered.
BCH code with a code length of 105 and a burst error correction code with 18 check symbols.

As mentioned above, the code C1 used in the present invention has a code length of 10
5. Since the number of test symbols is 14, the number of test symbols is smaller than any of the above conventional examples, and therefore the code C
It can be seen that 1 is more efficient.

In this way, when the correctable burst length is b, the code length in the conventional encoding/decoding system is nl(2b-1
), whereas in this embodiment.

Since it is given by approximately 2nl b, the ratio of the correctable burst length to the code length increases.

〔Effect of the invention〕

As explained above, according to the present invention, the ratio of the correctable burst length to the code length is relatively large, and burst errors with small weight can be corrected.
An extremely efficient encoding/decoding system can be provided.

[Brief explanation of the drawing]

FIG. 1 is a block diagram of an error correction encoding circuit showing an embodiment of the present invention, and FIG. 2 is a block diagram of an error correction decoding circuit. 101 to 114.501 to 514: Registers forming the feedback shift register, 301 to 307.7
01-708: Exclusive OR gate, 800: Pattern discriminator, 521-625: Reception information sequence shift register. Procedural amendment (voluntary) October 26, 1982

Claims (2)

[Claims] (1) The first generator polynomial of a cyclic code with two or more information symbols that has the ability to correct all random errors of t (t is any integer constant of 3 or more) or less, and the minimum distance (t+1
), a feedback shift register is configured using a product of a second generator polynomial of a cyclic code having a number of information symbols of 2 or more and a period different from a divisor of the period of the first generator polynomial as a generator polynomial; An error correction coding method characterized in that an information sequence is input to the feedback shift register to create an error correction cyclic code or a shortened cyclic code. (2) The first generator polynomial of a cyclic code with 2 or more information symbols that has the ability to correct all random errors of t (t: any integer constant of 3 or more) or less, and the minimum distance (t+1
), a feedback shift register is configured using a product of a second generator polynomial of a cyclic code having a number of information symbols of 2 or more and a period different from a divisor of the period of the first generator polynomial as a generator polynomial; The received information sequence is input to the feedback shift register, and at the same time input to the error correction buffer register. After calculating the syndrome in the feedback shift register, the generated syndrome is used to calculate the received information sequence. An error correction decoding method characterized by performing error correction.