CN109450458B - Method for determining linear block code performance boundary based on conditional three-error probability - Google Patents

Method for determining linear block code performance boundary based on conditional three-error probability Download PDF

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CN109450458B
CN109450458B CN201811605242.8A CN201811605242A CN109450458B CN 109450458 B CN109450458 B CN 109450458B CN 201811605242 A CN201811605242 A CN 201811605242A CN 109450458 B CN109450458 B CN 109450458B
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block code
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CN109450458A (en
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刘佳
戴红
安晓峰
吴丹
张明宇
王宇飞
矫利伟
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Jilin Teachers Institute of Engineering and Technology
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    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/13—Linear codes
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Abstract

The invention discloses a method for determining a linear block code performance boundary based on conditional three-error probability, which specifically comprises the following steps: step S1, the code word of the linear block code is modulated to become a sending signal, and the receiving end receives the receiving vector after the sending signal passes through an additive white Gaussian noise channel; step S2, calculating a probability density function; step S3, calculating the condition three error probabilities; step S4, calculating the optimal high-dimensional spherical radius; and step S5, obtaining the performance bound of the linear block code based on the conditional triple error probability, and further judging the error probability of the maximum likelihood decoding frame of the linear block code. The invention can effectively avoid the re-calculation of the decoding error probability of the repeated area, effectively reduce the repeated calculation of the decoding error probability area, effectively improve the existing spherical boundary by the determined linear block code performance boundary, tighten the linear block code performance boundary and reduce the error probability of the maximum likelihood decoding frame.

Description

Method for determining linear block code performance boundary based on conditional three-error probability
Technical Field
The invention belongs to the field of digital communication and digital storage, and particularly relates to a method for determining a linear block code performance boundary based on conditional triple error probability.
Background
In order to solve the problem of reliable and efficient communication in a complex scene, linear block codes suitable for various communication systems are continuously proposed, and how to examine the performance of one linear block code is particularly important. The decoding error probability of linear block codes can effectively measure the quality of the codes, however, in most cases, the decoding error probability of linear block codes cannot be described by an exact expression. Therefore, the decoding error probability is estimated mainly by means of Monte Carlo simulation and upper and lower bounds solution of a computer. The traditional method is to use computer monte carlo simulation to judge, but the method is time-consuming and energy-consuming, and firstly, a plurality of experiments are needed particularly for high-dimensional codes or high signal-to-noise ratios (SNRs), so that the computer continuously works for a plurality of days. Secondly, when the error rate is low, the bit decoding error probability cannot be obtained by Monte Carlo simulation. Thirdly, the Monte Carlo simulation result is difficult to intuitively describe the relationship between the error probability value and the system parameter, so that theoretical guidance on how to improve the system performance problem cannot be provided. For the defects of Monte Carlo simulation, the maximum likelihood decoding error probability upper and lower bound technology of the error correcting code is more concerned. For the upper bound techniques, in calculating the Frame-error rate (Frame-error rate) upper bound, it is generally only dependent on the weight spectrum of the code (or truncated weight spectrum), and no specific internal structure of the code needs to be obtained. The error probability upper bound technique can guide the parameter selection of the coding and decoding design on one hand, and can avoid time-consuming simulation evaluation on the other hand. In 2002, Sason indicates (variants on the Gallager boots, connections, and applications [ J ]. IEEE Transactions on Information Theory,2002,48(12): 3029-:
Figure BDA0001923439410000011
Wherein E represents an error event, Pr (E) represents a probability of the error event occurring,yis to receive the vector of the received data,
Figure BDA0001923439410000012
is a received vectoryFall in the area
Figure BDA0001923439410000021
The probability of the outer-most probability,
Figure BDA0001923439410000022
represents an arbitrary area around the point of transmission signal when
Figure BDA0001923439410000023
In time, the receiving end must decode with errors, therefore,
Figure BDA0001923439410000024
can be selected from
Figure BDA0001923439410000025
To obtain
Figure BDA0001923439410000026
When in use
Figure BDA0001923439410000027
The upper bound is performed by using a joint bound based on pair-wise error probability (PEP). In 1994, Herzberg and Poltyrev (Techniques of bounding the basic of decoding error for block coded modulation structures [ J]IEEE Transactions on Information Theory,1994,40(3): 903-. In 2018, an article entitled "in-depth study of binary linear block code sphere boundaries" published by liu jia et al, academy of agricultural engineering, sec happy, details that the sphere boundaries proposed by Kasami et al are equivalent to the sphere boundaries SB proposed by Herzberg and Poltyrev. In 2016, Zhao et al proposed nested Gallager regions
Figure BDA0001923439410000028
(Sphere bound revisited:A new simulation approach to performance evaluation of binary linear codes over AWGN channels[C]// Proceeding of International Symposium on Turbo Codes and Iterative Information Processing, Brest, France,2016: 330-:
Figure BDA0001923439410000029
wherein, fu(r) represents probability
Figure BDA00019234394100000210
A calculable upper bound of (a) is,
Figure BDA00019234394100000211
indicating area
Figure BDA00019234394100000212
G (r) represents a random variable
Figure BDA00019234394100000213
Is determined. Zhao et al propose conditional pairwise error probabilities and re-derive the spherical bounds. Since the existing sphere boundaries are all based on pairwise error probabilities, it is inevitable to cause repeated calculation of error probability regions. The repeated calculation of error probability regions is one of the main reasons for the lack of tight sphere boundaries. The current method avoids or reduces the repeated calculation of error probability region by reducing the number of code words or reducing the number of terms of the joint boundary, and the famous scholars Agrell indicates (On the Voronoi neighbor ratio for binary linear codes, "IEEE Transactions On Information Theory,1998,44: 3064-. The solution of the local weight spectrum needs to analyze the position relation of all code words through the theory of combinatorics. Currently, there are few codes that can solve the local weight spectrum, such as random linear codes, hamming codes, partial BCH codes, and Reed Muller codes, which, while tightening the upper bound to some extent, solve the decoded local weight spectrum as a non-deterministic polynomial Hard (NP-Hard) problem.
Disclosure of Invention
The invention aims to provide a method for determining a linear block code performance boundary based on conditional triple error probability, which solves the problems that in the prior art, a spherical boundary is not compact and repeated calculation of an error probability area exists, the existing computer Monte Carlo simulation method is time-consuming and energy-consuming, especially for high-dimensional codes or high signal-to-noise ratios, the calculation complexity is high, and when the error rate is low, the error probability of bit decoding cannot be obtained by Monte Carlo simulation.
The technical scheme adopted by the invention is that the method for determining the performance boundary of the linear block code based on the conditional three-error probability specifically comprises the following steps:
step S1, definition of transmission signal and reception vector:
the linear block code is
Figure BDA0001923439410000031
ct∈{0，1}，0≤t≤n-1，
Figure BDA0001923439410000032
Is binary linear block code, where k is information length and n is code length of linear block code, and is converted into transmission signal after binary phase shift keying BPSK modulations＝(s0,s1,…st,…sn-1) Wherein s ist＝1-2ctT is more than or equal to 0 and less than or equal to n-1, and the transmitting end sends a signals＝(s0,s1,…st,…sn-1) The vector is transmitted to an additive white Gaussian noise channel (AWGN), and the receiving vector received by a receiving end is
Figure BDA0001923439410000033
Wherein the content of the first and second substances,zthe representative channel noise is a vector formed by n independent uniformly distributed random variables, the obedience mean value of each random variable is 0, and the variance is sigma 2(ii) a gaussian distribution of;
step S2, calculating a probability density function g (r);
step S3, calculating the condition three error probability p3(r,i,j)；
Step S4, calculating the optimal high-dimensional spherical radius r1；
Step S5, obtaining the performance bound of linear block code based on conditional triple error probability
Figure BDA0001923439410000034
And then the error probability of the maximum likelihood decoding frame of the linear block code is judged.
Further, the step S2 is specifically performed according to the following steps:
the all-zero code word is obtained from the step S1c (0)After binary phase shift keying BPSK modulation, the vector (0,0, …,0) becomes a transmission signal vectors (0)The nesting area selected by the performance boundary of linear block code is the vector of the transmitted signal (1,1, …,1)s (0)An n-dimensional sphere with r ≥ 0 as radius, i.e. sphere center
Figure BDA0001923439410000035
Probability density function of n-dimensional spherical radius r
Figure BDA0001923439410000036
Where σ represents channel noisezEach random variable in (a) is subject to a standard deviation in a normal distribution,
Figure BDA0001923439410000041
Figure BDA0001923439410000042
wherein t represents any real variable,
Figure BDA0001923439410000043
represents a plurality of numbers
Figure BDA0001923439410000044
The real part value of (a);
the linear block code has geometric uniformity, which means that the geometric distribution of all codewords in the euclidean space is symmetrical and uniform, and the error probability caused by sending any codeword is equal according to the maximum likelihood criterion, so that it is assumed that all-zero codewords are transmitted in the channel.
Further, the step S3 is specifically performed according to the following steps:
selecting a fixed codewordc (1)As reference code word, a fixed code word is assumedc (1)Has a Hamming weight of d1＝WH(c (1)) ≧ 1, wherein the function WH(c) Representing code words of a linear block codecDefined as i being a code word of a linear block codecAnd all-zero code wordc (0)Hamming distance therebetween, i.e., W ═ iH(c-c (0)) (ii) a Defining j as a code word of a linear block codecAnd fix the code wordc (1)Hamming distance therebetween, i.e., j ═ WH(c-c (1)) (ii) a Definition of
Figure BDA0001923439410000045
For binary linear block codes
Figure BDA00019234394100000412
Wherein X is a first dummy variable, Y is a second dummy variable, and a triangular spectrum Bi,j(c (1)) Is a code word of a linear block codecNumber of (2), triangle spectrum Bi,j(c (1)) And fixed code wordc (1)Correlation, when context is clear, fixed codewords are omittedc (1)Definition of Bi,jIs a triangular spectrum of binary linear block codes, wherein i is more than or equal to 0, j is more than or equal to n,s (1)is a fixed code wordc (1)Defining the transmitted signal vector obtained by BPSK modulation
Figure BDA0001923439410000046
Let p be3(r, i, j) is shown at event
Figure BDA0001923439410000047
Maximum likelihood decoding under occurrence conditions into three error probabilities
Figure BDA0001923439410000048
Wherein the content of the first and second substances,
Figure BDA0001923439410000049
denotes an n-dimensional sphere of radius r, f: (y) Representing a received vectoryPr { } denotes the probability;
receiving a vectoryUniformly distributed on n-dimensional spherical surface with radius of r
Figure BDA00019234394100000410
Conditional to three error probabilities
Figure BDA00019234394100000411
As shown in (b) of FIG. 1, on a spherical surface
Figure BDA0001923439410000051
The conditional-to-three error probability of (c) is obtained from the ratio of the geometric body surface area formed by the intersection of two high-dimensional spherical caps, i.e. the dotted line portion, to the surface area of the high-dimensional sphere, i.e.,
Figure BDA0001923439410000052
wherein, d1Is a fixed code wordc (1)The weight of the Hamming that is,
Figure BDA0001923439410000053
Figure BDA0001923439410000054
further, the step S4 is specifically performed according to the following steps:
fu(r) is a conditional union bound, i.e.
Figure BDA0001923439410000055
Wherein, Bi,j(c (1)) Is a binary linear block code
Figure BDA0001923439410000056
Triangular spectrum of (1), p2(r,d1) Is the conditional pairwise error probability;
conditional pairwise error probability p2(r,d1) As shown in formula (7):
Figure BDA0001923439410000057
as shown in FIG. 1 (a), the conditional pairwise error probability p2(r,d1) The surface area ratio of a high-dimensional spherical crown, namely the dotted line part, to the high-dimensional sphere is obtained;
Figure BDA0001923439410000058
due to p3(r, i, j) is a non-decreasing continuous function with respect to the n-dimensional spherical radius r, and p3(0, i, j) ═ 0 and p3(+ ∞, i, j) — (pi + theta)/2 pi, where phi is a variable representing an angle and theta is a vector
Figure BDA0001923439410000061
Sum vector
Figure BDA0001923439410000062
Angle formed, therefore, the union of conditions bound fu(r) is also a non-decreasing continuous function with respect to the n-dimensional spherical radius r, and fu(0) 0 and fu(+ ∞) is not less than 1, and fu(r) in the interval
Figure BDA0001923439410000063
Is strictly increasing, and
Figure BDA0001923439410000064
there is only one optimal high-dimensional spherical radius r1Satisfy the requirement of
Figure BDA0001923439410000065
Wherein i ═ WH(c-c (0))，j＝WH(c-c (1))，p2(r1,d1) At a radius r1Sent by a sending terminal on the high-dimensional spherical surfacec (0)Receiving end error decoding into fixed code wordc (1)Conditional pairwise error probability of (c); p is a radical of3(r1I, j) is at a radius r1Sent by a sending terminal on the high-dimensional spherical surfacec (0)Receiving end error decoding into fixed code wordc (1)And linear block code codewordcIs a three error probability.
The invention has the advantages that the invention researches the space distribution rule of the code words, prevents or reduces the repeated calculation of the error probability area, finds out the repeated area which is easy to make errors in decoding by researching the space distribution rule of any three code words, can effectively avoid the repeated calculation of the decoding error probability of the repeated area, and effectively reduces the repeated calculation of the decoding error probability area. The invention provides a compact linear block code performance boundary which can be effectively used for judging the performance of all linear block codes and is not influenced by any condition.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a geometric graph of the conditional pairwise error probability and the conditional triplex error probability of the present invention;
FIG. 2 is a comparison curve of the original spherical boundary and the linear block code performance boundary of the Hamming code maximum likelihood decoding frame error probability in accordance with the embodiment 1 of the present invention;
FIG. 3 is a block diagram of convolutional code encoding applied in embodiment 2 of the present invention;
fig. 4 is a comparison curve of the original spherical boundary and the linear block code performance boundary of the maximum likelihood decoding frame error probability of the convolutional code in embodiment 2 of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Example 1
The selected binary linear block code is [7, 4 ]]The code length n of the linear block code is 7, and the information length k is 4. Selecting a fixed code word with a minimum weight of 3c (1)Is a reference code word, i.e. WH(c (1))＝d 13, for calculating the triangular spectrum. By calculation, the triangle enumeration function is B (X, Y) ═ Y 3+X3+6X3X4+6X4Y3+X4Y7+X7X4To obtain Bi,j(c (1)) The linear block code performance bound determined by the invention based on the conditional three error probabilities is utilized to obtain the linear block code performance bound:
(1) at a given point
Figure BDA0001923439410000071
In which EbIs the power of the signal, N0Is the power spectral density of the noise
Figure BDA0001923439410000072
Go through r ∈ (0, infinity), find
Figure BDA0001923439410000073
Wherein n is 7;
(2) joint boundary of conditions
Figure BDA0001923439410000074
Wherein k is 4, n is 7, d1Go through r ∈ (0, infinity) to find 3
Figure BDA0001923439410000075
And
Figure BDA0001923439410000081
(3) by
Figure BDA0001923439410000082
Obtaining the optimal high-dimensional spherical radius r1；
By calculation, g (r), fu(r)、r1And substituting the linear block code performance range into the formula (10).
The calculation result is shown in fig. 2, and as can be seen from fig. 2, the performance boundary of the linear block code provided by the invention is more compact, and the quality of the linear block code can be more effectively evaluated, the invention firstly calculates a triangular enumeration function by researching the spatial distribution rule of any three code words, secondly finds out a repeat region with the error of the maximum likelihood decoding, reduces the repeat calculation of an error probability region, effectively avoids the re-calculation of the decoding error probability of the repeat region, and finally achieves the purposes of tightening the performance boundary of the linear block code and reducing the error probability of a maximum likelihood decoding frame.
Example 2
The selected binary linear block code is [204, 100 ] ]A convolutional code, where the code length is n-204 and the information length is k-100, and its encoder outputs two bits v through the encoder every 1 bit u as shown in fig. 3(0)v(1)Code rate of
Figure BDA0001923439410000083
An encoder is a four-state device that generates a polynomial g (D) ═ 1+ D2,1+D+D2]Wherein g (D) represents a generator polynomial of a convolutional code, and D represents a delay operator. Selecting a fixed code word with a minimum weight of 5c (1)Is a reference code word, i.e. WH(c (1))＝d 15, for calculating a triangular spectrum, only the path in the first error event is considered when calculating the triangular spectrum. The linear block code performance boundary is obtained by utilizing the linear block code performance boundary determined by the invention based on the conditional three-error probability, and the specific calculation process is as follows:
(1) at a given point
Figure BDA0001923439410000084
Where SNR is the signal-to-noise ratio, EbIs the power of the signal, N0Is the power spectral density of the noise, is obtained
Figure BDA0001923439410000091
Go through r ∈ (0, infinity), find
Figure BDA0001923439410000092
Wherein, n is 204;
(2) joint boundary of conditions
Figure BDA0001923439410000093
Wherein k is 100, n is 204, d1Go through r e (0, infinity) to find 5
Figure BDA0001923439410000094
And
Figure BDA0001923439410000095
(3) by
Figure BDA0001923439410000096
Obtaining the optimal high-dimensional spherical radius r1；
By calculation, g (r), fu(r)、r1And substituting the linear block code performance range into the formula (10).
The calculation result is shown in figure 4, and as can be seen from figure 4, the linear block code performance boundary in the invention is tighter than the original spherical boundary, the invention firstly calculates the triangular enumeration function by researching the spatial distribution rule of any three code words, secondly finds out the repeat region with the error of the maximum likelihood decoding, reduces the repeat calculation of the error probability region, effectively avoids the re-calculation of the decoding error probability of the repeat region, and finally achieves the purposes of tightening the linear block code performance boundary and reducing the error probability of the maximum likelihood decoding frame.
The linear block code performance bound of the invention is more compact than the upper bound of the spherical bound proposed by Zhao et al, and the invention calculates fuAnd (r) finding out a repeat region with decoding errors by researching the spatial distribution rule of any three code words, effectively avoiding the recalculation of the decoding error probability of the repeat region, and providing the method for determining the performance boundary of the linear block code based on the conditional triple error probability. Thirdly, the linear block code performance bound based on the conditional triple error probability provided by the invention is an exact mathematical expression, the upper bound of the maximum likelihood decoding error probability of the linear block code can be quickly obtained through calculation, meanwhile, the formula (10) plays a theoretical guidance role in coding design, and the defects of time consumption and energy consumption of computer Monte Carlo simulation are overcome: 1. in studying the performance of certain codes, e.g., maximum likelihood decoding such as RS, Turbo, and LDPC codes, or the performance of codes at high SNR, the time complexity of Monte Carlo simulations can be very highFor example, maximum likelihood decoding of RS codes is a difficult problem of non-deterministic polynomials and cannot be achieved by computer monte carlo simulation. 2. Even though some error correction codes can be judged to be good or bad by means of Monte Carlo simulation, the Monte Carlo simulation consumes energy. 3. The Monte Carlo simulation results are difficult to intuitively describe the relationship between the error probability value and the system parameters, so that theoretical guidance on how to improve the system performance cannot be provided. Therefore, the present invention can be rapidly used to judge the quality of the linear block code.
All the embodiments in the present specification are described in a related manner, and the same and similar parts among the embodiments may be referred to each other, and each embodiment focuses on differences from other embodiments. In particular, for the system embodiment, since it is substantially similar to the method embodiment, the description is simple, and for the relevant points, reference may be made to the partial description of the method embodiment.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (1)

1. A method for determining a performance bound of a linear block code based on a conditional triplet error probability is characterized by comprising the following steps:
step S1, definition of transmission signal and reception vector:
the linear block code is
Figure FDA0003615511400000011
ct∈{0，1}，0≤t≤n-1，
Figure FDA0003615511400000012
Is binary linear block code, where k is information length and n is code length of linear block code, and is converted into transmission signal after binary phase shift keying BPSK modulations＝(s0,s1,…st,…sn-1) Wherein s ist＝1-2ctT is more than or equal to 0 and less than or equal to n-1, and the transmitting end sends a signals＝(s0,s1,…st,…sn-1) The vector is transmitted to an additive white Gaussian noise channel (AWGN), and the receiving vector received by a receiving end is y＝s+z＝(y0,y1,…yt,…yn-1)∈RnWherein, in the step (A),zrepresenting the channel noise as a vector consisting of n independent and identically distributed random variables, wherein the obedient mean value of each random variable is 0, and the variance is sigma2(ii) a gaussian distribution of;
step S2, calculating a probability density function g (r);
step S3, calculating the condition three error probability p3(r,i,j)；
Step S4, calculating the optimal high-dimensional spherical radius r1；
Step S5, obtaining the performance bound of linear block code based on conditional triple error probability
Figure FDA0003615511400000013
Further judging the error probability of the maximum likelihood decoding frame of the linear block code;
the step S2 is specifically performed according to the following steps:
the all-zero code word is obtained from the step S1c (0)After binary phase shift keying BPSK modulation, the vector (0,0, …,0) becomes a transmission signal vectors (0)The nesting area selected by the performance boundary of linear block code is the vector of the transmitted signal (1,1, …,1)s (0)An n-dimensional sphere with r ≥ 0 as radius, i.e. sphere center
Figure FDA0003615511400000014
Probability density function of n-dimensional spherical radius r
Figure FDA0003615511400000015
Where σ represents channel noisezEach random variable in (a) is subject to a standard deviation in a normal distribution,
Figure FDA0003615511400000016
Figure FDA0003615511400000017
wherein t represents any real variable,
Figure FDA0003615511400000018
represents a plurality of numbers
Figure FDA0003615511400000019
The real part value of (a);
the step S3 is specifically performed according to the following steps:
selecting a fixed codewordc (1)As reference code word, a fixed code word is assumedc (1)The Hamming weight of is d 1＝WH(c (1)) ≧ 1, wherein, the function WH(c) Representing code words of a linear block codecDefined as i being a code word of a linear block codecAnd all-zero code wordc (0)Hamming distance therebetween, i.e., W ═ iH(c-c (0)) (ii) a Defining j as a code word of a linear block codecAnd fix the code wordc (1)Hamming distance therebetween, i.e., j ═ WH(c-c (1)) (ii) a Definition of
Figure FDA0003615511400000021
For binary linear block codes
Figure FDA0003615511400000022
Wherein X is a first dummy variable, Y is a second dummy variable, and a triangular spectrum Bi,j(c (1)) Is a code word of a linear block codecNumber of (2), triangle spectrum Bi,j(c (1)) And fixed code wordc (1)Correlation, when context is clear, fixed codewords are omittedc (1)Definition of Bi,jIs a triangular spectrum of binary linear block codes, wherein i is more than or equal to 0, j is more than or equal to n,s (1)is a fixed code wordc (1)Defining the transmitted signal vector obtained by BPSK modulation
Figure FDA0003615511400000023
Let p be3(r, i, j) is shown at event
Figure FDA0003615511400000024
Maximum likelihood decoding under occurrence conditions into three error probabilities
Figure FDA0003615511400000025
Wherein the content of the first and second substances,
Figure FDA0003615511400000026
denotes an n-dimensional sphere of radius r, f: (y) Representing a received vectoryPr { } denotes the probability;
receiving a vectoryUniformly distributed on n-dimensional spherical surface with radius of r
Figure FDA0003615511400000027
Conditional to three error probabilities
Figure FDA0003615511400000028
On the spherical surface
Figure FDA0003615511400000029
The conditional three-dimensional error probability of (1) is obtained by the surface area of a geometric body formed by intersecting two high-dimensional spherical crowns, namely the surface area ratio of a dotted line part to a high-dimensional sphere, namely,
Figure FDA0003615511400000031
Wherein, d1Is a fixed code wordc (1)The weight of the Hamming that is,
Figure FDA0003615511400000032
Figure FDA0003615511400000033
v represents an integral variable;
the step S4 is specifically performed according to the following steps:
fu(r) is a conditional union bound, i.e.
Figure FDA0003615511400000034
Wherein, Bi,j(c (1)) Is a binary linear block code
Figure FDA0003615511400000035
Triangular spectrum of (1), p2(r,d1) Is the conditional pairwise error probability;
conditional pairwise error probability p2(r,d1) As shown in the following formula:
Figure FDA0003615511400000036
conditional pairwise error probability p2(r,d1) The surface area ratio of a high-dimensional spherical crown, namely the dotted line part, to the high-dimensional sphere is obtained;
Figure FDA0003615511400000037
where φ is a variable representing an angle;
there is only one optimal high-dimensional spherical radius r1Satisfy the requirement of
Figure FDA0003615511400000038
Wherein i ═ WH(c-c (0))，j＝WH(c-c (1))，p2(r1,d1) At a radius r1Sent by a sending terminal on the high-dimensional spherical surfacec (0)Receiving end error decoding into fixed code wordc (1)Conditional pairwise error probability of (c); p is a radical of3(r1I, j) is at a radius r1The sending end on the high-dimensional spherical surface sendsc (0)Receiving end error decoding into fixed code wordc (1)And linear block code codewordcIs a three error probability.
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