Coding method, decoding method, coder, and decoder

Abstract

An encoding method of generating an encoded sequence by performing encoding of a given encoding rate based on a predetermined parity check matrix. The predetermined matrix is either a first parity check matrix or a second parity check matrix. The first parity check matrix corresponds to a low density parity check (LDPC) convolutional code that uses a plurality of parity check polynomials, and the second parity check matrix is generated by performing at least one of row permutation and column permutation on the first parity check matrix. A parity check polynomial satisfying zero of the LDPC convolutional code is expressible by using a specific mathematical expression.

Claims

1. An encoding method comprising: generating, by performing encoding of coding rate 3/5 on an information sequence X.sub.1, an information sequence X.sub.2, and an information sequence X.sub.3, an encoded sequence composed of the information sequence X.sub.1, the information sequence X.sub.2, the information sequence X.sub.3, and a parity sequence P, the encoding based on a predetermined parity check matrix having mz rows and 2mz columns, where m is an even number no smaller than two and z is a natural number, wherein the predetermined parity check matrix is one of a first parity check matrix and a second parity check matrix, the first parity check matrix corresponding to a low density parity check (LDPC) convolutional code that uses a plurality of parity check polynomials, the second parity check matrix being generated by performing at least one of row permutation and column permutation on the first parity check matrix, wherein two parity check polynomials satisfying zero are provided for each of 1P.sub.1(D) and 1P.sub.2(D) in accordance with the LDPC convolutional code, wherein each parity check polynomial satisfying zero of the LDPC convolutional code is expressed by one of a first group of expressions or one of a second group of expressions, wherein the first group of expressions consist of the following expressions: $\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),2}+1}{\overset{r_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),3}+1}{\overset{r_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),0}{P}_2\left(D\right)=\left(D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{r}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{r}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),0}{P}_2\left(D\right)=0;$ $\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),2}+1}{\overset{r_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),3}+1}{\overset{r_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),1}{P}_1\left(D\right)=\left(D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{r}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{r}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),1}{P}_1\left(D\right)=0;$ $\left(\underset{s=R_{\#\left(2i\right),1}+1}{\overset{r_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),2}{P}_1\left(D\right)=\left(D^{\#\left(2i\right),1,}{r}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,1}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),2}{P}_1\left(D\right)=0;$ $\left(\underset{s=R_{\#\left(2i\right),1}+1}{\overset{r_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),3}{P}_2\left(D\right)=\left(D^{\#\left(2i\right),1,}{r}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,1}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),3}{P}_2\left(D\right)=0$ where p is an integer no smaller than one and no greater than three, q is an integer no smaller than one and no greater than r.sub.#(2i),p, when r.sub.#(2),p is a natural number, X.sub.p(D) is a polynomial expression of the information sequence X.sub.p and P(D) is a polynomial expression of the parity sequence P, D being a delay operator, a.sub.#(2i),p,q and .sub.#(2i),0 are natural numbers, .sub.#(2i),1 is a natural number, .sub.#(2i),2 is an integer no smaller than zero, .sub.#(2i),3 is a natural number, R.sub.#(2i),p is a natural number, 1R.sub.#(2i),p<r.sub.#(2i),p holds true, and .sub.#(2i),p,y.sub.#(2i),p,z holds true for .sup.(y, z) where y is an integer no smaller than one and no greater than r.sub.#(2i),p, z is an integer no smaller than one and no greater than r.sub.#2i,p, and y and z satisfy yz, and wherein the second group of expressions consist of the following expressions: $\left(\underset{s=R_{\#\left(2i+1\right),1}+1}{\overset{r_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),0}{P}_2\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{r}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,1}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),0}{P}_2\left(D\right)=0;$ $\left(\underset{s=R_{\#\left(2i+1\right),1}+1}{\overset{r_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),1}{P}_1\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{r}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,1}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),1}{P}_1\left(D\right)=0;$ $\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),2}+1}{\overset{r_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),3}+1}{\overset{r_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),2}{P}_1\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{r}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{r}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),2}{P}_1\left(D\right)=0;$ $\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),2}+1}{\overset{r_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),3}+1}{\overset{r_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),2}{P}_2\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{r}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{r}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),3}{P}_2\left(D\right)=0$ where p is an integer no smaller than one and no greater than three, q is an integer no smaller than one and no greater than r.sub.#(2i+1),p, when r.sub.#(2i+1),p is a natural number, X.sub.p(D) is a polynomial expression of the information sequence X.sub.p and P(D) is a polynomial expression of the parity sequence P, D being a delay operator, .sub.#(2i+1),p,q and #(.sub.2i+1),0 are natural numbers, .sub.#(2i+1),1 is a natural number, .sub.#(2i+1),2 is an integer no smaller than zero, .sub.#(2i+1),3 is a natural number, R.sub.#(2i+1),p is a natural number, 1R.sub.#(2i+1),p<r.sub.#(2i+1),p holds true, and .sub.#(2i+1),p,y.sub.#(2i+1),p,z holds true for .sup.(y, z) where y is an integer no smaller than one and no greater than r.sub.#(2i+1),p, z is an integer no smaller than one and no greater than r.sub.(#2i+1),p, and y and z satisfy yz.

2. A decoding method of decoding an encoded sequence encoded by employing a predetermined encoding method, wherein the predetermined encoding method generates the encoded sequence by performing encoding of coding rate 3/5 on an information sequence X.sub.1, an information sequence X.sub.2, and an information sequence X.sub.3, the encoded sequence composed of the information sequence X.sub.1, the information sequence X.sub.2, the information sequence X.sub.3, and a parity sequence P, the encoding based on a predetermined parity check matrix having mz rows and 2mz columns, where m is an even number no smaller than two and z is a natural number, wherein the predetermined parity check matrix is one of a first parity check matrix and a second parity check matrix, the first parity check matrix corresponding to a low density parity check (LDPC) convolutional code that uses a plurality of parity check polynomials, the second parity check matrix being generated by performing at least one of row permutation and column permutation on the first parity check matrix, wherein two parity check polynomials satisfying zero are provided for each of 1P.sub.1(D) and 1P.sub.2(D) in accordance with the LDPC convolutional code, wherein each parity check polynomial satisfying zero of the LDPC convolutional code is expressed by one of a first group of expressions or one of a second group of expressions, wherein the first group of expressions consist of the following expressions: $\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),2}+1}{\overset{r_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),3}+1}{\overset{r_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),0}{P}_2\left(D\right)=\left(D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{r}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{r}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,}{R}_{\#\left(2i,\right),3}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),0}{P}_2\left(D\right)=0;$ $\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),2}+1}{\overset{r_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i\right),3}+1}{\overset{r_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),1}{P}_1\left(D\right)=\left(D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{r}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{r}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i\right),1}{P}_1\left(D\right)=0;$ $\left(\underset{s=R_{\#\left(2i\right),2}+1}{\overset{r_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),2}{P}_1\left(D\right)=\left(D^{\#\left(2i\right),1,}{r}_{\#\left(2i\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1}{R}_{\#\left(2i\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,1}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),2}{P}_1\left(D\right)=0;$ $\left(\underset{s=R_{\#\left(2i\right),1}+1}{\overset{r_{\#\left(2i\right),1}}{.Math.}}{D}^{\#\left(2i\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),2}}{.Math.}}{D}^{\#\left(2i\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i\right),3}}{.Math.}}{D}^{\#\left(2i\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),3}{P}_2\left(D\right)=\left(D^{\#\left(2i\right),1,}{r}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i\right),1,}{R}_{\#\left(2i\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i\right),2,}{R}_{\#\left(2i\right),2}+\mathrm{.Math.}+D^{\#\left(2i\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i\right),3,}{R}_{\#\left(2i\right),3}+\mathrm{.Math.}+D^{\#\left(2i\right),3,1}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i\right),3}{P}_2\left(D\right)=0$ where p is an integer no smaller than one and no greater than three, q is an integer no smaller than one and no greater than r.sub.#(2i),p, when r.sub.#(2i),p is a natural number, X.sub.p(D) is a polynomial expression of the information sequence X.sub.p and P(D) is a polynomial expression of the parity sequence P, D being a delay operator, .sub.#(2i),p,q and .sub.#(2i),0 are natural numbers, .sub.#(2i),1 is a natural number, .sub.#(2i),2 is an integer no smaller than zero, .sub.#(2i),3 is a natural number, R.sub.#(2i),p is a natural number, 1R.sub.#(2),p<r.sub.#(2),p holds true, and .sub.#(2i),p,y.sub.#(2i),p,z holds true for .sup.(y, z) where y is an integer no smaller than one and no greater than r.sub.#(2i),p, z is an integer no smaller than one and no greater than r.sub.#2i,p, and y and z satisfy yz, wherein the second group of expressions consist of the following expressions: $\left(\underset{s=R_{\#\left(2i+1\right),1}+1}{\overset{r_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),0}{P}_2\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{r}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,1}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),0}{P}_2\left(D\right)=0;$ $\left(\underset{s=R_{\#\left(2i+1\right),1}+1}{\overset{r_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),1}{P}_1\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{r}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,1}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,1}+1\right)X_3\left(D\right)+P_1\left(D\right)+D^{\#\left(2i+1\right),1}{P}_1\left(D\right)=0;$ $\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),2}+1}{\overset{r_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),3}+1}{\overset{r_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),2}{P}_1\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{r}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{r}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),2}{P}_1\left(D\right)=0;$ $\left(1+\underset{s=1}{\overset{R_{\#\left(2i+1\right),1}}{.Math.}}{D}^{\#\left(2i+1\right),1,s}\right)X_1\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),2}+1}{\overset{r_{\#\left(2i+1\right),2}}{.Math.}}{D}^{\#\left(2i+1\right),2,s}\right)X_2\left(D\right)+\left(\underset{s=R_{\#\left(2i+1\right),3}+1}{\overset{r_{\#\left(2i+1\right),3}}{.Math.}}{D}^{\#\left(2i+1\right),3,s}\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),2}{P}_2\left(D\right)=\left(D^{\#\left(2i+1\right),1,}{R}_{\#\left(2i+1\right),1}+\mathrm{.Math.}+D^{\#\left(2i+1\right),1,1}+1\right)X_1\left(D\right)+\left(D^{\#\left(2i+1\right),2,}{r}_{\#\left(2i+1\right),2}+\mathrm{.Math.}+D^{\#\left(2i+1\right),2,}{R}_{\#\left(2i+1\right),2}+1\right)X_2\left(D\right)+\left(D^{\#\left(2i+1\right),3,}{r}_{\#\left(2i+1\right),3}+\mathrm{.Math.}+D^{\#\left(2i+1\right),3,}{R}_{\#\left(2i+1\right),3}+1\right)X_3\left(D\right)+P_2\left(D\right)+D^{\#\left(2i+1\right),3}{P}_2\left(D\right)=0$ where p is an integer no smaller than one and no greater than three, q is an integer no smaller than one and no greater than r.sub.#(2i+1),p, when r.sub.#(2i+1),p is a natural number, X.sub.p(D) is a polynomial expression of the information sequence X.sub.p and P(D) is a polynomial expression of the parity sequence P, D being a delay operator, .sub.#(2i+1),p,q and .sub.#(2i+1),0 are natural numbers, .sub.#(2i+1),1 is a natural number, .sub.#(2i+1),2 is an integer no smaller than zero, .sub.#(2i+1),3 is a natural number, R.sub.#(2i+1),p is a natural number, 1R.sub.#(2i+1),p<r.sub.#(2i+1),p holds true, and a.sub.#(2i+1),p,y.sub.#(2i+1),p,z holds true for .sup.(y, z) where y is an integer no smaller than one and no greater than r.sub.#(2i+1),p, z is an integer no smaller than one and no greater than r.sub.(#2i+1),p, and y and z satisfy yz, and wherein the decoding method comprises decoding the encoded sequence based on the predetermined parity check matrix and by using belief propagation (BP).