JPH1098394A - Method for decoding grating code - Google Patents

Abstract

PROBLEM TO BE SOLVED: To decode a grating code with a wide free distance by using a plural ity of processors, so as to decode the code according to a specific procedure. SOLUTION: A processor P<(> L<)> decides a set TM<(> L<)> including 2<m> of distance sets with respect to 2<m> of binary sets V<(> L<)> (t+Λ<(1)> +...+Λ<(> L<-1)> ), based on a signal y(t+i) to be decoded and gives the result to a processor P<(> L<-1)> , where Λ<(1)> ,..., Λ<(> L<)> are constant and not negative. The processor P<(1)> decides 2<m> sets of TM<(1)> , including the distance with respect to 2<m> sets of possible values v<(1)> and gives the result to 2<m> sets of TM<(1)> , where (l) is L-1, L-2,..., 1. The processor P<(0)> recovers converted signals u, v<(0)> and the recovered signal v<(0)> is fed back into the processor P<(1)> . The processor P<(1)> feeds back V<(l+1)> to the processor P<(l+1)> , where (l)=1, 2,..., L-1.

Description

Description: BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a grid code decoding method. 2. Description of the Related Art In a digital communication system, information transmission is disturbed by channel defects such as channel noise and the like, and transmission errors are likely to occur. Digital communication systems that require high reliability require channel coding that is less likely to cause transmission errors. In channel coding design, the digital information corresponds to the codeword or
Maps to a code path. A set of all codewords is called a code. The distance characteristic between codewords is effective for correcting a communication error and improves communication performance. The mapping between the information set and the codeword set is determined by coding (Cod
ing) or encoding (Encoding)
g). If the signal of each codeword is a binary signal, the channel coding is binary coding. Mapping is sometimes called code. A procedure for restoring information from a received signal that may be disturbed by an error is called decoding. [0003] Binary trellis codes are a commonly used coding technique. For a k / n binary lattice code,
In each time unit, when k communication bits are input to the encoder, n code bits are output. The n code bits are affected not only by the k communication bits used in the current time unit as inputs to the encoder, but also by the communication bits used as inputs in earlier time units. A codeword of a binary lattice code is represented by a lattice path. The most important binary lattice code is a binary convolutional code (Binary c
onvolutional code). Binary convolutional codes are linear, time-invariant binary lattice codes, introduced several decades ago and still in widespread use. A paper entitled "Channel Coding Using Multi-Level / Phase Signals"("Channel code
ing with multilevel / phase
signals, "IEEE Trans. Info.
rm. Theory. , Vol 28, no. 1, p. 5
5-67, 1982), G. Ungerboeck has proposed a new channel coding called trellis code modulation (TCM) that combines trellis coding and modulation. Signal space Ω
Are 2 m signal points {z ₁ , z ₂ ,..., Z
_2m }. z∈ ｛z ₁ , z ₂ , ...,
z _2m } and S ₁ , S ₂ ,..., S _m {0, 1}, each signal point in Ω has its own m When designing a TCM with a number of information bits, Ungerboeck's TCM encoding is as shown in FIG. In the t-th time unit, folding t)). [0005] Binary grid codes and grid code modulation (TC
M) can be classified into lattice codes. The performance of the lattice code is mainly evaluated by three variables: coding rate, decoding difficulty, and decoding error rate. The goal is to design a lattice code with a high coding rate, easy decoding, and a low decoding error rate.
To reduce the decoding error rate of the lattice coding system, a free distance (Free distant) is required.
ce) needs to be widened. [0006] Lin and Wang are 1
U.S. patent application no. 08 /
398, 797, a multilevel delay processor (Multilevel delay p) between a binary convolutional encoder and a signal mapper (Signal mapper).
A grid code for encoding by introducing a processor is proposed. Figure 2 shows this encoding.
Shown in ) And sent to the multi-level delay processor. Multi-level delay processor t)) is obtained as the final output. Decoding of a lattice code using this lattice encoding method can be performed by using a lattice of C. SUMMARY OF THE INVENTION In the present invention, a new type of lattice code is designed which modifies the above method to encode by introducing more multi-level delay processors and signal mappers. . An object of the present invention is to provide a grid code having a large free distance and to provide a decoding method therefor. SUMMARY OF THE INVENTION The grid code coding method according to the present invention starts with a binary grid code C ^(1). Therefore, this is a grid code T decoding method that can perform encoding. Where y (t + i) is the received signal to be decoded, Λ ⁽¹⁾ ,..., Λ ^(L) is a non-negative constant,
λ is the truncation length used for decoding C ⁽¹⁾ , and is the (t + Λ ⁽¹⁾ +... + Λ ^(L) ) th time unit, and the set ｛y (t + i) .. + Λ ^(L−1) ), a set T _M ^(L) (t + Λ ⁽¹⁾ +... + Λ ^(L−1) ) including 2 ^m distances is determined, and a set of distances T _M ^(L) (t + Λ) is determined. ⁽¹⁾ + ... + Λ
^(L-1) ) to the processor P ^(L-1) , and for 1 = L-1, L-2,.
｛T _M ^{(l + 1)} (t + i): i ≦ Λ ⁽¹⁾ T _M ⁽¹⁾ (t + Λ ⁾ containing 2 ^m distances to
⁽¹⁾ +... + Λ ^(l−1) ), where Λ ⁽¹⁾ is a non-negative constant and the set of distances T _M ⁽¹⁾ (t + Λ ^)
(1) +... + Λ ^(l−1) ) to the processor P ^(l−1)
(B) supplying to Is set to the Viterbi decoding algorithm processing on the lattice code C ⁽¹⁾ . ⁾ And (t-λ + 1) a step of feeding back to the processor ^{P (1) (c),} (D). [0010] ^{_{, V 2 (1) (t}} ), ..., v m (1) (t)) into a can be replaced by the encoder of the plurality of binary trellis code, the L multilevel delay processor and the L signal The Viterbi decoding algorithm for C ⁽¹⁾ processed by the mapper and used in processor P ⁽⁰⁾ is replaced by the Viterbi decoding algorithm for the binary lattice codes. [0011] In addition, r-bit communication is performed using an encoder for ^one binary grid signal C ^{(1). 1)} (t) = (v ₁ ⁽¹⁾ (t), v ₂ ⁽¹⁾ (t),
, V _m ⁽¹⁾ (t)) and processing the output of the binary grid signal through a multi-level delay processor Q ⁽¹ ) to obtain λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ , …, Λ _m
⁽¹⁾ is a non-negative constant, step ii), which can represent t)); Selecting step iii); Λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ _m ⁽¹⁾ are non-negative constants, (T) is a binary set of m elements, and if l = L, v ^{(l + 1)} (t) is in the signal space Ω, and the output of the signal point, step iv), gives Encoding may be performed. [0012] Better. The binary lattice code or the plurality of binary lattice codes may be a linear time-invariant code or a plurality of linear time-invariant codes, that is, a binary convolutional code or a plurality of binary convolutional codes. . The λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ _m ⁽¹⁾ can be different values and / or λ ₁ ⁽¹⁾ , λ
₂ ⁽¹⁾ ,..., Λ _m ⁽¹⁾ can be non-zero. Λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ _m-1 ⁽¹⁾
Are ideally all constants λ ⁽¹⁾ , and λ _m ⁽¹⁾ is preferably zero. The signal space Ω is an array of signals, which can be used for decoding lattice code modulation. The signal space Ω is m
It is represented by a set of binary sets of elements and can be used for coding in a binary grid code. It is represented by a collection, and m is a positive integer. M / m of the set can be decomposed into ⁽¹⁾ sets,
Where m ⁽¹⁾ is a positive integer Where m ^{(L + 1)} is a positive integer. Preferred embodiments of the present invention will be described below in detail with reference to preferred embodiments. These examples are:
First, a large free distance
e) it is possible to design a lattice code using
It shows that encoding can be performed using the multi-level encoding method described in FIG. And these examples demonstrate the potential of the decoding method of the present invention. Signal space (Signal space) Ω
Are 2 ^m signal points (Signalpoint) z ₁ ,
z _2, ..., and it consists of _{z 2m.} The signal space Ω is z∈
{Z ₁ , z ₂ ,..., Z _2m } and s ₁ , s _{2 ,.}
For {0,1}, signal point z is a unique m-set of binary numbers (m The level distance _{ｐ p} of the space Ω is defined as follows. If Ω is an array of signals, then △ (z,
z ′) represents the square Euclidean distance between z and z ′. That is, D ² (z, z ′). Then, if Ω is an array of m-sets of binary numbers (sets of m elements), ｚ (z, z ′) represents a Hamming distance between z and z. That is,
d (z, z ′). The distance structure of the signal space is
It can be said that { ₁ , △ ₂ , ..., _ｍm }. For example, the 8PSK signal array is divided into a three-level structure as shown in FIG. In contrast, the structural distance is _{１ 1}
= D ₁ ² = 0.586, △ ₂ = D ₂ ² = 2, _{３ 3} = D ₃
² = 4. Further, for example, a binary 2-set (set of two elements) Ω = {0,1} ² = ｛z ₀ = (0,
0), z ₁ = (1, 0), z ₂ = (0, 1), z ₃ =
(1,1) is divided as follows. The distance structure for Ω is And (Each is an r-set.) Thus, 1 <
For l <L + 1, ), Which indicates that the code is a binary lattice code. C ^{(L + 1)} also indicates that it is a desirable lattice code T. In general, the free distance of C ^{(l + 1} ) is l = 1,2, ... L
From the distance characteristic of C ^{(l + 1)} with respect to. Below,
C ^(L) from ^{C (L + 1)} = derivation of the free distance T is described, where the delay constant And [0018] ), ...} is the congruent s-sequence obtained through the multi-level delay processor Q ^(L) Let d _{p be} in the p-th level among the d locations that are different. This state is there. [0019] Represents the square Euclidean distance between. Ω is an m-set of binary numbers (consisting of m elements Indicates that [0020] When, Given by [0021] As described in the example, if the signal space is {0, 1} ² ,
It will have a distance structure of ｛△ ₁ = d ₁ = 1, △ ₂ = d ₂ ² = 2｝. For i = 1,2, d Make it fruit. [0022] Can be proved. Here, d (v, v ') =
(D ₁ , d ₂ ,..., D _m ). If Ω is a signal set, △ _free is T
CMC ^{(l + 1)} = square free distance T. Ω is an m-set of binary numbers (set consisting of m elements)
Then △ _free is the binary grid code C
It shows that _{(L + 1)} = T free distance. To decode the lattice code T, a lattice is used for the binary convolutional code C ⁽¹⁾ . For C ⁽¹⁾ , if the number of encoder memory bits is ν, the number of states of the congruent lattice (Number of st
ate) is 2 ^ν . As described below, when L = 2, it is used to represent a decoding procedure. In that procedure, the lag constant is 1 ≦ l ≦ L = 2, And Let ⁽¹⁾ denote the total delay between the first and last level of the first multi-level delay processor. That is, for l = 1 and 2, Λ ⁽¹⁾ =
(M-1) λ ⁽¹⁾ . Binary convolutional code C
Let λ be the length at which the vertex used to decode ⁽¹⁾ is cut at the level. λ ⁽¹⁾ ≧ λ, λ ⁽²⁾ ≧
λ + Λ ⁽¹⁾ Is accepted. A decoding procedure with ≧ λ is shown as follows. Step 1 p = 1, 2,..., M, v _p ⁽²⁾ (t + (1)) = 0 and v _p ⁽²⁾ (t + (1)) = 1, the bit distance M
_vp ⁽²⁾ _{(t + Λ} ⁽¹⁾ ₎ is calculated by the following equation. ^{(P-i) λ (2} ) -Λ (1) ≧ λ _So, ^{v i (2)}
(T-((pi) λ ⁽²⁾ -Λ ⁽¹⁾ )) has already been recovered, and Note that Further, _{s i} is _s ^{i (2)} in the above equations in order to simplify the notation ^{(t + Λ (1) +} Λ
⁽²⁾ -Used to replace (p-1) (2)). Mv ₁ ⁽²⁾ _{(t + Λ} ⁽¹⁾ ₎ ,..., M _vm
⁽²⁾ By summing _{(t + Λ} ⁽¹⁾ ₎ , the distance M
_v ^{(2 (1)} ,..., _Vm ⁽²⁾ (t + Λ ⁽¹⁾ )). Distance T _M ⁽²⁾ (t + Λ ⁽¹⁾ ) = ｛M _{v (} Used. Step 2 For p = 1, 2,..., M, the bit distance M _vp ⁽¹⁾
_(T) Calculate by Where ^{_{t p = t + Λ (1}} ) - (p
−l) λ ⁽¹⁾ and s _i ⁽¹⁾ were recovered by assumption as i <p. M _v1 ⁽¹⁾ _(t) ,..., M _vm ⁽¹⁾
By summing _(t) , the distance M _v ⁽¹⁾ _(t) { ^M } is used in step 3. Step 3 ) Applies the Viterbi algorithm to the 2 ^ν state lattice that becomes C ^(1), and if i ≧ 0, the distance I do. The decoding procedure then returns to step 1. Note that λ Λ ⁽¹⁾ + Λ ⁽²⁾ ) is recovered at the タ イ ム th time unit. λ + Λ ⁽¹⁾ +... + Λ ^(L) ) Recover at the ⁽ th) time unit. There, for each l, Λ ^(L) ≧ Σ
_{i =} 1λ ⁽¹⁾ , λ ⁽¹⁾ ≧ (λ + Λ ⁽¹⁾ +... + Λ
^(L-1) ). Now, consider the first embodiment. Let L = 2,
Select Ω, which is an 8PSK signal group. It is ｛Δ ₁ = 0.5
86, Δ ₂ = 2, Δ ₃ = 4｝. The delay constant is 1 ≦ l ≦ L, And Mapping w for the first signal mapper ⁽¹⁾
(S ₁ ⁽¹⁾ , s ₂ ⁽¹⁾ , s ₃ ⁽¹⁾ ) = (v ₁
⁽²⁾ , v ₂ ⁽²⁾ , s ₃ ⁽²⁾ ) are as follows. A 2/3 rate binary convolutional code C ⁽¹⁾ with V = 2 is used, where the number of lattice states is 2 ²
= 4. The generator matrix of code C ⁽¹⁾ is It is. As described above, the lattice code T is a TCM whose code rate is 2 information bits per signal point in an 8PSK signal group. From the computer search, the free distance of this TCM squared is _Δfree
= D _free ² ≥ 7.516. Compared to uncoded QPSK, this TCM is AWG
It has a near line code gain of 5.75 dB in N channels. In 1982, Wangerboeck reported that the code gain of the near-short line was 3.0d for the 4-state TCM considered.
It's just B. Using the decoding method of the present invention,
The simulation results obtained by setting the delay constants λ = λ ⁽¹⁾ = 30 and λ ⁽¹⁾ = 90 are shown in FIG. From FIG. 9, using the decoding method of the present invention for this TCM, the code gain over uncoded QPSK is about 3.6 dB with a bit error rate of 10 ⁻⁶ . Consider a second embodiment where L is also equal to two. Choose Ω = {0,1} ² which is divided into a two-level structure having a distance structure of {Δ ₁ = 1, Δ ₂ = 2}.
The delay constant is 1 ≦ l ≦ L = 2, And Both mappings of the signal mapper are defined the same as: Here, w = w ⁽¹⁾ , w ⁽²⁾
It is. A 率 rate binary code C with ν = 2
^{Using (1)} , it includes that the number of lattice states is 2 ² = 4. The generator matrix of C ⁽¹⁾ is G =
(57). Thus, the lattice code T is a binary lattice code having a code rate of 1/2. From the computer search, the free distance of this binary lattice code is calculated to be at least 11. In comparison,
The most famous binary convolutional code with four lattice states has only 5 free distances. Simulation results obtained by using the decoding method of the present invention and setting λ = λ ⁽¹⁾ = 20 and λ ⁽²⁾ = 40 are shown in FIG.
Indicated by From FIG. 10, it can be seen that a bit error rate of 10 ⁻⁶ is obtained with E _b / N _o = 5.2 dB. [0033] In each of the two embodiments described above, the first delay processor lambda _p ⁽¹⁾ is p∈ {1, ..., m- 1} and relative l = 1, 2 lambda ⁽¹⁾ Are considered equal. A third embodiment is provided to illustrate the more general case of T. Then, some of the delay times λ _p ⁽¹⁾ , _p {(1,..., M− ^{1) of} the ^first multi-level processor are set to 0. The third embodiment is different from the first embodiment. The signal space Ω used in the third embodiment, the first multilevel delay processor Q ⁽¹⁾ , and the signal mappers S ⁽¹⁾ and S ⁽²⁾ used in the first embodiment are used. However, in the third embodiment, the truncation length λ and the delay constant λ _p ⁽¹⁾ are λ =
λ ₁ ⁽¹⁾ = λ ₂ ⁽¹⁾ = 30, λ ₃ ⁽¹⁾ = λ ₂
⁽²⁾ = λ ₃ ⁽²⁾ = 0 and λ ₁ ⁽²⁾ = 90 are set. Further, T is a TCM, free distance squared This can be calculated to be ^{_{Δ free = D free 2 ≧ 5.76}} . The decoding of the third embodiment is similar to the decoding of the first embodiment except for minor changes. In the third embodiment, the decoding delay time is shorter than that of the first embodiment. Simulation results obtained using the decoding of the present invention are also shown in FIG. Error of the third embodiment
The performance is almost as good as the error performance of the first embodiment. One of the features of the first multi-level delay processor used for encoding T shown in FIG. 5 is to process the input bits at each level so that the p-th level is ( It is delayed by λ _p ⁽¹⁾ time unit compared to the ⁽ p + 1) th level. In general, l, λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ , ...
Each of λ _m ⁽¹⁾ need not necessarily be equal, and some of λ _p ⁽¹⁾ can be set to zero. Basically, this encoding uses a binary convolutional code encoder to generate m bits, which are then encoded by L multi-level delay processors and L signal mappers at L ≧ 2. To process. This encoding can easily be generalized to a multi-level encoding based on an encoder of several binary convolutional codes. The design is as follows. Q (q <
m) Using the encoder, V ₁ ⁽¹⁾ (t), v _{2
^{(1) (t), ...}} v m (1) (t) and m bits generated completely such, they further L (L ≧ 2) was treated by the multi-level delayed processor and the signal mapper, the signal space Ω Select a signal point. In this case, the proposed decoding method is slightly modified to
⁽⁰⁾ applies the Viterbi decoding algorithm to several binary lattice codes instead of applying the Viterbi decoding algorithm to a single binary lattice code. Further, there is a method for generalizing the lattice code T which can be executed in the following manner, so as to be suitable for the decoding method of the present invention. Each time unit To generate a binary set of m / m ⁽¹⁾ m ⁽¹⁾ elements combined to represent ^{m (L + 1)} ) In Ω consisting of signal points, m / ^L
m ^{(l + 1)} signal points. Therefore, in each time unit, the lattice code T is output as Ω
M ^{) (T)} represents a binary set of several m ⁽¹⁾ factors in the range of 1 ≦ l ≦ L, to show that representative of the several signal points in Ω at l = L + 1 range, still It is valid. Finally, the encoder of the binary convolutional code C ⁽¹⁾ used for encoding and decoding of T can be replaced by a more general binary lattice code encoder.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 Ungerboeck
FIG. 4 is a diagram illustrating an encoding method for TCM. FIG. 2 is a diagram illustrating an encoding method for a lattice code proposed by Lin and Wan. FIG. 3 is a diagram illustrating an encoding method of a lattice code T suitable for the decoding method of the present invention. FIG. 4 is a diagram illustrating a decoding method of a lattice code T according to the present invention. FIG. 5 is a diagram illustrating a multilevel stage encoding method of the lattice code T according to the present invention. FIG. 6 is a functional diagram of an l-th multilevel delay processor according to the present invention. FIG. 7 is a functional diagram of an l-th signal mapper according to the present invention. FIG. 8 is a diagram showing an 8PSK signal array in the present invention. FIG. 9 is a diagram illustrating a TCM simulation result in an additional white Gaussian noise channel according to the first and third embodiments of the present invention. FIG. 10 is a diagram illustrating a simulation result of a binary lattice code in an additional white Gaussian noise channel according to the second embodiment of the present invention.

Claims (1)

Claims 1. First, using an encoder of a binary grid code C ⁽¹⁾ , A decoding method of a trellis code T, wherein y (t + i) is the received signal to be decoded,
Λ ⁽¹⁾ ,..., Λ ^(L) is a non-negative constant, λ is the truncation length used for decoding C ⁽¹⁾ ,
In the (t + Λ ⁽¹⁾ +... + ^{） (L)} ) th time unit, the set ｛y (t + i .. + Λ ^(L−1) ), a set T _M ^(L) (t + Λ ⁽¹⁾ +... + Λ ^(L−1) ) including 2 ^m distances is determined, and a set of distances T _M ^(L) (t + Λ) is determined. ( ¹⁾ + ... + Λ
^(L-1) ) to the processor P ^(L-1) (a), and for 1 = L-1, L-2 _,.
^{(L + 1)} (t + i): i ≦ Λ ⁽¹⁾ T _M ⁽¹⁾ (t + Λ ⁾ containing ^{2 m} distances to
⁽¹⁾ +... + Λ ^(l−1) ), where Λ ⁽¹⁾ is a non-negative constant and the set of distances T _M ⁽¹⁾ (t + Λ ^)
(1) +... + Λ ^(l−1) ) to the processor P ^(l−1)
(B) supplying to Is set to the Viterbi decoding algorithm processing on the lattice code C ⁽¹⁾ . ⁾ And (t-λ + 1) a step of feeding back to the processor ^{P (1) (c),} (D). A lattice code decoding method, comprising: _{^{= (V 1 (1) (}} t), v 2 (1) (t), ..., v m
⁽¹⁾ It can be replaced by a plurality of binary lattice code encoders that convert to (t)), processed by L multi-level delay processors and L signal mappers, and used by processor P ⁽⁰⁾ The decoding method according to claim 1, wherein the Viterbi decoding algorithm for C ⁽¹⁾ is replaced with a plurality of Viterbi decoding algorithms for the plurality of binary lattice codes. 3. An r-bit encoder using a ^single binary grid signal C ⁽¹⁾ encoder. Obtaining i), processing the output of the binary grid signal through a multi-level delay processor Q ⁽¹⁾ , and obtaining λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,.
λ _m ⁽¹⁾ is a non-negative constant, step ii), which can represent t)); Selecting step iii); Λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ _m ⁽¹⁾ are non-negative constants, 3. The decoding method according to claim 1, wherein the encoding of the lattice code T is performed by step iv), which is in the middle Ω and is the output of the signal point. 4. The parameter の of the processor P ^(1)
(1) is 1 ≦ l ≦ L Law. 5. The binary lattice code or the plurality of binary lattice codes is a linear non-time-varying code or a plurality of linear non-time-varying codes, that is, a binary convolutional code or a plurality of binary convolutional codes. The decoding method according to claim 1 or 2, wherein: 6. The λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ _m
4. The method according to claim 3, wherein ⁽¹⁾ can be different values and / or λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ _m ⁽¹⁾ are not zero. Coding method. 7. The λ ₁ ⁽¹⁾ , λ ₂ ⁽¹⁾ ,..., Λ
_m-1 ⁽¹⁾ are ideally all constants λ ⁽¹⁾ ,
The decoding method according to claim 3, wherein λ _m ⁽¹⁾ is 0. 8. The decoding method according to claim 1, wherein the signal space Ω is an array of signals, and can be used for decoding of lattice code modulation. 9. The decoding method according to claim 1, wherein the signal space Ω is represented by a set of binary numbers of m elements and can be used for coding with a binary lattice code. 3. The decoding method according to claim 1, wherein a total number of m / m ⁽¹⁾ is represented by m, and m is a positive integer. m ⁽¹⁾ can be decomposed into m / m ⁽¹⁾ sets of binary sets of elements, where ^M / m ^{(L + 1)} signal points, and m
^{2. The method according} to claim 1, wherein ^{(L + 1)} is a positive integer.
Or the decoding method according to 2.