Optimal local repair code construction method based on projective plane mid-arc

Abstract

The invention belongs to the technical field of distributed storage. The invention provides an optimal local repair code construction method based on arcs in a projective plane, and the method comprises the steps: taking out # imgabs1 # points from the arcs # imgabs0 #, dividing each # imgabs2 # points into a group to form a point set, and obtaining # imgabs3 # point sets; each parameter of the check matrix is determined according to the property of points in each point set, for # imgabs4, a submatrix # imgabs6 # of a # imgabs5 local repair group of the check matrix exactly forms a standard generation matrix with the parameter of # imgabs7 # MDS code, and the optimal local repair code with the parameter of # imgabs8 #, the minimum distance # imgabs9 # and the disjoint repair group with the parameter of # imgabs10 # is generated. According to the structure, when a plurality of errors occur in the group, a small number of survival nodes can be used for repairing and expanding the range of the code length # imgabs11 # to # imgabs12 #, so that the code rate is improved.

Description

Technical Field

[0001] The present invention relates to the field of distributed storage technology, and specifically relates to a method for constructing an optimal locally repairable code based on an arc in a projective plane, a method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane, a computer device, a computer-readable storage medium, and a computer program product. Background Technique

[0002] The statements in this part only provide background techniques related to the present invention and do not necessarily constitute prior art.

[0003] The big data era has brought a sharp increase in the scale of data. How to store, access, and repair massive data more efficiently and at lower cost is a key issue in the storage field. In recent years, distributed storage has become the main solution to this problem. It adopts an extensible system structure and uses multiple storage servers to share the storage load. It has now been widely applied in commercial practices, such as Microsoft's cloud storage project Azure Storage, Amazon's Dynamo, and Google's storage system Colossus, etc. Generally speaking, in a distributed storage system, a certain amount of redundant data is usually generated through a data redundancy mechanism to ensure the reliability of the system. The two most common redundant data maintenance technologies are replication and erasure codes respectively.

[0004] Replication is a relatively traditional data redundancy technology, that is, redundancy is generated by creating copies. As long as the copy files are available, the original files will not be lost. However, the storage overhead of this replication strategy is very large. To reduce the storage overhead, the concept of erasure codes emerged. When an erasure code with parameters is used to store data, first the data is divided into segments, and then redundant segments are added. The resulting segments are stored in nodes. Therefore, the storage overhead generated is given by . In addition, erasure codes can effectively handle the failure of a single storage node in a distributed storage system. When a single node fails, the system will reconstruct the entire data file by contacting the auxiliary nodes and downloading the data in the auxiliary nodes. Currently, distributed storage systems mainly adopt two types of coding methods, locally repairable codes and regenerating codes, to cope with the challenges of node repair.

[0005] The concept of local repairability was first proposed by Gopalan et al. For a code , if the -th coordinate component (codeword symbol) of the codeword can be repaired by no more than other coordinate components (codeword symbols), then the codeword coordinate component With local repairability If all coordinate components (codeword symbols) in have local repairability, then is called a locally repairable code (LRC) with local repairability . Since is much smaller than the original data size, the hard disk I / O time consumption can be greatly reduced during repair. A linear code with local repairability is denoted as -locally repairable code, briefly denoted as -LRC, where represents the code length, represents the dimension, represents the minimum distance, and the LRC with local repairability only repairs a single node.

[0006] To repair multiple faulty nodes, Prakash et al. introduced LRC with locality. Let be an -ary linear code. For the -th coordinate component of the codeword in code , if there exists a subset ofsuch that , and , then the -th coordinate component is said to have locality, where represents the size of the set , represents restricted to the minimum distance of the corresponding code. If each coordinate component of the codeword in has locality, then is said to have locality. Such a code is denoted as -ary -LRC. It can be seen from the definition that -LRC is equivalent to -LRC. The minimum distance of the code with parameters -LRC satisfies the inequality: If the equality holds, it is called a Singleton-optimal -LRC.

[0007] From the Singleton - type bound, for a given minimum distance and local repairability and parameters of the Singleton - optimal locally repairable code (LRC), the larger the code length means the larger the code rate. Therefore, in the case of given , , and , constructing a Singleton - optimal -LRC with a larger code length has always been a hot research issue in this field.

[0008] Let be a prime power, denote the finite field with elements. Let be an integer, and a -arc in the finite projective plane is a set consisting of

[0009] points, where no three points are collinear. When is odd, a -arc in the finite projective plane

[0010] is called an oval. ; The point set in the finite projective plane forms an oval in

[0011] When is even, a -arc in the finite projective plane is called a hyperoval. Let and be a power of 2. A construction of a hyperoval is as follows: ; Currently, there is a construction that can obtain a code with length , minimum distance and locality , satisfying and and having disjoint repair groups -ary Singleton optimal LRC, but this construction can only correspond to fixed parameters in the case of, and is only applicable to the LRC for repairing a single node within a repair group, with a relatively small application scope, and the code length is restricted to . SUMMARY OF THE INVENTION

[0012] To solve the deficiencies of the prior art, the present invention provides an optimal local repair code construction method based on arcs in a projective plane, constructing an optimal one with parameters , and this construction can repair multiple faulty nodes within a group and expand the code length range to and improve the code rate. In this construction, each local repair group contains nodes, and ensures that each node can be quickly repaired by at most surviving nodes within the local group when damaged, while tolerating at most nodes being damaged simultaneously. This construction reaches the Singleton bound and achieves high fault tolerance with limited storage redundancy, especially suitable for large-scale distributed storage scenarios. To achieve the above object, the present invention adopts the following technical solutions:

[0013] In the first aspect, the present invention provides an optimal local repair code construction method based on arcs in a projective plane. An optimal local repair code construction method based on arcs in a projective plane includes the following process:

[0014] Let represent the maximum number of faulty nodes that can be tolerated during the local repair process; let be a prime power and satisfy . Let be a positive integer that satisfies , and when is odd, ; when is even, ; when .

[0015] When is odd, let the point set be an oval in the projective plane ; when is even, let the point set be a hyperoval in the projective plane ; take points from the point set , and divide every points into a group to form a point set, obtaining point sets; Determine each parameter of the parity-check matrix according to the properties of the points in each point set. For , the -th sub-matrix of the local repair group of the parity-check matrix exactly constitutes a standard generator matrix of an MDS code with parameters , having a locality of , a minimum distance of , and generating an optimal locally repairable code with disjoint repair groups and parameters .

[0016] As a further limitation of the first aspect of the present invention, the general form of the parity-check matrix is:

[0017] where is an -order identity matrix, and represents the all-zero matrix in . For , each is a column vector in , and each is a column vector in . It is easy to verify that for an optimal locally repairable code with disjoint repair groups and parameters , and in the case where the number of repair groups , ; As a further limitation of the first aspect of the present invention, when is odd, ; when is even, ; where is a -element finite field.

[0018] As a further limitation of the first aspect of the present invention, take points from the point set , and divide every points into a group to form a point set, obtaining point sets, denoted as , and each point set is expressed as: ; ; …; ; For any , the point set satisfies: , and any three points in the set are not collinear.

[0019] As a further limitation of the first aspect of the present invention, according to the obtained point set , for , , calculate the dual subspace of each point in , let , and obtain line sets ; For , each line set has lines, that is . According to the duality principle of the projective plane, for any , the line set satisfies: , and any three lines in the set do not intersect at the same point; For , take three lines in , let , , where , and respectively represent the one-dimensional spaces spanned by the non-zero vectors in , , , that is, 3 points in the projective plane ; The vectors , and obtained on this basis are used as the lower half vectors of the th repair group in the parity-check matrix ; for , let and ; Since any three lines in do not intersect at the same point, so are all non-zero elements in, and then the upper half parameters in the parity-check matrix are: , , ; Among them, .

[0020] In a second aspect, the present invention provides a method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane.

[0021] A method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane includes the following processes: Based on the method for constructing an optimal locally repairable code using an arc in a projective plane described in the first aspect of the present invention, a parity-check matrix of the locally repairable code is given ; Let denote the linear code with as the parity-check matrix. For any codeword in , divide its coordinate components into groups of every components, and a total of groups are obtained; For , multiply the -th submatrix in the parity-check matrix and the -th group of coordinate components of any codeword in the linear code to obtain the corresponding parity-check equation, and obtain the repair scheme for each failed node in the -th group according to the parity-check equation.

[0022] According to the properties of the submatrix , it can be verified that this locally repairable code has locality.

[0023] In a third aspect, a computer device is provided, which is characterized by including: a processor and a computer-readable storage medium; the processor is adapted to execute a computer program; the computer-readable storage medium stores a computer program, and when the computer program is executed by the processor, the method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane as described in the second aspect of the present invention is implemented.

[0024] In a fourth aspect, the present invention provides a computer-readable storage medium that stores a computer program, and the computer program is adapted to be loaded and executed by a processor to implement the method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane as described in the second aspect of the present invention.

[0025] In a fifth aspect, the present invention provides a computer program product that includes a computer program, and when the computer program is executed by a processor, the method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane as described in the second aspect of the present invention is implemented.

[0026] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The present invention innovatively proposes an optimal local repair code (LRC) generation strategy based on arcs in the projective plane. The core of this strategy lies in designing a special parity-check matrix structure by using the properties of arcs in the projective plane and the duality principle. Each sub-matrix exactly constitutes a standard generator matrix of an MDS (Maximum Distance Separable) code. This structure ensures that each local repair group has a local degree of; combined with the lower half vectors of the th repair block, and , which further enables the minimum distance of this local repair code to reach . Through this ingenious design, the present invention constructs an optimal local repair code with non-overlapping repair groups, which can achieve efficient repair when multiple faulty nodes occur within a group, significantly enhancing the fault tolerance of the code.

[0027] 2. The optimal local repair code designed by the present invention has a local degree of, where the parameter represents the number of nodes required for local repair, and the value of is relatively small. Specifically, when multiple node failures occur within a group, the system only needs to access fewer surviving nodes to complete the repair operation, without having to access the entire storage system. This design significantly reduces the bandwidth overhead and computational complexity during the repair process, thus greatly improving the repair efficiency. In addition, the parameter represents the fault tolerance of local repair, that is, within a local group, at most nodes can fail simultaneously without affecting the repairability of the data. By reasonably optimizing the values of and , the present invention further reduces the storage redundancy while ensuring data reliability and repair efficiency, achieving efficient utilization of storage resources.

[0028] 3. By optimizing the structure of the local repair code, the present invention significantly improves the performance and applicability of the code. Specifically, this strategy expands the range of the code length to , which not only breaks through the limitations of traditional local repair codes but also further improves the code rate, enabling in a finite field Under it, more efficient storage and data recovery can be achieved. This extended code length range provides greater flexibility for distributed storage systems, especially when facing large-scale data storage and high reliability requirements. It can significantly reduce storage costs and improve the overall performance of the system. In addition, the disjoint repair group design ensures that data can still be quickly recovered when multiple nodes fail simultaneously, further enhancing the robustness of the system.

[0029] Advantages of additional aspects of the present invention will be given in part in the following description, become apparent in part from the following description, or be learned through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The accompanying drawings forming a part of this specification are used to provide a further understanding of the present invention. The schematic embodiments and descriptions thereof of the present invention are used to explain the present invention and do not constitute an improper limitation to the present invention.

[0031] Figure 1 It is a schematic flow chart of a method for constructing an optimal locally repairable code based on an arc in a projective plane provided for Embodiment 1 of the present invention; Figure 2 It is a schematic flow chart of repairing a failed node of an optimal locally repairable code based on an arc in a projective plane provided for Embodiment 5 of the present invention; Figure 3 It is a schematic diagram of a computer device provided for Embodiment 6 of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0032] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0033] It should be noted that the following detailed descriptions are all exemplary and are intended to provide further explanations of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0034] In the case of no conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0035] Embodiment 1: This implementation proposes a method for constructing an optimal locally repairable code based on an arc in a projective plane, and constructs an optimal locally repairable code with parameters which can achieve repair when multiple failed nodes occur within a group, and expand the code length range to and improve the code rate. As shown in Figure 1 , the specific construction method includes the following process: Step 1: Let represent the maximum number of failed nodes that can be tolerated during the local repair process; assume is a prime power and satisfies . Let be a positive integer that satisfies , and when is odd, ; when is even, .

[0036] When is odd, let be defined by formula (1) and form an oval in the projective plane .

[0037] (1) When is even, let be defined by formula (2) and form a hyperoval in the projective plane .

[0038] (2) Take points from the point set and divide every points into a group to form a point set, obtaining point sets, denoted as , and each set can be expressed as: (3); (4); ··· (5); It is easy to check that for any , the point set satisfies the following conditions: Condition 1: ; Condition 2: Any three points in the set are not collinear; this is determined by the properties of the oval and hyperoval.

[0039] Step 2: According to the point set obtained in Step 1, for , , calculate (the dual subspace of each point in ), and let , then line sets can be obtained. For , each line set has lines, such that (6); It is easy to check that for any , the line set satisfies the following conditions: Condition 1: ; Condition 2: Any three lines in the set do not intersect at the same point. This is obtained according to the duality principle of the projective plane.

[0040] For , take three lines in , and let: , , (7); where , and respectively represent the one-dimensional spaces spanned by the non-zero vectors in , , , that is, 3 points in the projective plane . Based on this, the vectors , and are used as the lower half vectors of the th repair group in the parity-check matrix . And, the three lines can be expressed as 。

[0041] For the remaining lines in the set, that is , because intersects with and , and is spanned by ; is spanned by , let: , (8); Since any three lines in it do not intersect at the same point, so are all non-zero elements in , so (9); Step 3: Construct the parity-check matrix The upper half submatrix of the th repair group in (10); Among them, , let be the all-1 column vector of length on ; The column vector, from top to bottom elements are used as the parameter in formula (9) of step 2, where .

[0042] Furthermore, the column vector, where . For , the specific form of the column vector is as follows: (11); (12); (13); Step 4: From , , obtained in step 2 and vector obtained in step 3, the parity-check matrix of the optimal locally repairable code with parameters can be obtained: (14); Among them, is the order identity matrix, and represents the all-zero matrix in .

[0043] The parity-check matrix satisfies that each submatrix exactly constitutes a MDS code standard generator matrix, where . Thus, an optimal locally repairable code with a code length of , a dimension of , a locality of , and a minimum distance of with non-overlapping repair groups is obtained.

[0044] Example 2: In the case of = 11, an example of constructing an optimal locally repairable code with parameters includes: Step 1: Let = 2, = 11, because is odd, so according to formula (1), all points are taken from the point set as { (1,7,5), (0,0,1), (1,5,3), (1,8,9), (1,0,0), (1,2,4), (1,3,9), (1,10,1), (1,4,5), (1,1,1), (1,9,4), (1,6,3)}. Then these points are divided into point sets. Each set is represented as: ; ; ; Step 2: Calculate according to the point sets obtained in Step 1 and formulas (6), (7), (8): = T , = T , = T ; = T , = T , = T ; = T , = T , = T ; Step 3: Calculate , , to get: = (1), = (10), = (10); = (1), = (5), =(6); =(1), =(2), =(2); Step 4: From the , , obtained in Step 2 and the vector obtained in Step 3, the following parity-check matrix can be obtained:

[0045] Thus, the following construction example of the optimal locally repairable code with parameters is obtained.

[0046] Example 3: In the case of = 9, a construction example of the optimal locally repairable code with parameters includes: Step 1: Let = 3, = 9, and let be a primitive element of satisfying its minimal polynomial . Since is odd, according to Equation (1), all q + 1 points are taken from the point set as { (1, 2, 1), (1, , ), (1, , 2), (1, , ), (1, 0, 0), (0, 0, 1), (1, 1, 1), (1, , ), (1, , 2), (1, , )}. Then these n = q + 1 points are divided into point sets. Each set is expressed as: ; ; Step 2: According to the point sets obtained in Step 1 and Formulas (6), (7), and (8), calculate: = T , = T , = T ; = T , = T , = T ; Step 3: According to Step 2 and formulas (11), (12), (13), calculate , , to obtain: =(1,1) T , =( , ) T , =( , ) T ; =(1,1) T , =(2, ) T , =(2, ) T ; Step 4: From the , , obtained in Step 2 and the vector obtained in Step 3, the following parity-check matrix can be obtained:

[0047] Thus, the construction example of the optimal locally repairable code with the above parameters is obtained.

[0048] Example 4: In the case of = 8, the construction example of the optimal locally repairable code with parameters includes: Step 1: Let = 3, = 8, and let be the primitive element of satisfying its minimal polynomial as . Since is even, according to Equation (2), from the point set Take out all q + 2 points from it as { (1, , ), (0, 0, 1), (1, , ), (1, , ), (1, , ), (1, , ), (1, , ), (0, , ), (1, , ), (1, , )}. Then divide these n = q + 2 points into point sets. Each set is represented as: ; ; Step 2: According to the point sets obtained in Step 1 and formulas (6), (7), (8), calculate: = T , = T , = T ; = T , = T , = T ; Step 3: According to Step 2 and formulas (11), (12), (13), calculate , , to get: = (1, 1) T , = ([[]] , ) T , = (1,[[[]] ) T ; = (1, 1)T , =( , 1) T , =( , ) T ; Step 4: From the , , obtained in Step 2 and the vector obtained in Step 3, the following parity-check matrix can be obtained:

[0049] Thus, an example of constructing an optimal locally repairable code with the above parameters is obtained. Example 5:

[0050] This implementation provides a method for repairing faulty nodes of an optimal locally repairable code based on an arc in a projective plane. As shown, the specific repair method includes the following process: Figure 2 Step 1: Based on the method for constructing an optimal locally repairable code using an arc in a projective plane described in Example 1, the following form of parity-check matrix is obtained: :

[0051] Step 2: Assume that the linear code uses as the parity-check matrix. For any codeword in , divide the coordinate components of into groups of every components, and a total of groups are obtained, that is, let: , , , (15); Step 3: Use to obtain the parity-check equation. For , the parity-check equation obtained by multiplying the th submatrix in the parity-check matrix and the th group of coordinate components is: (16); where , and thus the Repair solutions for each faulty node.

[0052] Since the submatrix exactly forms a standard generator matrix of an MDS code and at the same time forms a parity-check matrix of the MDS code, it can be verified that this locally repairable code has locality. For example, when the first nodes in the th group all fail, according to formula (16), it can be known that these faulty nodes can all be repaired (linearly represented) by the th, , and rd nodes in the

[0053] Example 6: As Figure 3 shown, this implementation provides an electronic device, which includes a processor 1001, a communication interface 1002, and a computer-readable storage medium 1003. Among them, the processor 1001, the communication interface 1002, and the computer-readable storage medium 1003 can be connected through a bus or other means.

[0054] Among them, the communication interface 1002 is used to receive and send data. The computer-readable storage medium 1003 can be stored in the memory of the electronic device. The computer-readable storage medium 1003 is used to store a computer program. The computer program includes program instructions. The processor 1001 is used to execute the program instructions stored in the computer-readable storage medium 1003.

[0055] The processor 1001 (or CPU (Central Processing Unit, central processor)) is the computing core and control core of the electronic device, and is suitable for implementing one or more instructions. Specifically, it is suitable for loading and executing one or more instructions to implement the corresponding method flow or corresponding function.

[0056] The processor 1001 is configured to execute the following process: Based on the method for constructing the parity-check matrix of the optimal locally repairable code using arcs in the projective plane described in Embodiment 1, give the parity-check matrix of this locally repairable code ; Let represent the linear code with as the parity-check matrix. For any codeword in , divide its coordinate components into groups of every components, and a total of For , the parity-check matrix in the sub - matrix and a linear code any codeword of the group coordinate components are multiplied to obtain the corresponding parity - check equation, and according to the parity - check equation, the repair scheme of each failed node in the group is obtained.

[0057] Example 7: This implementation provides a computer - readable storage medium (Memory). A computer - readable storage medium is a memory device in an electronic device for storing programs and data. It can be understood that the computer - readable storage medium here can include both the built - in storage medium in the electronic device and, of course, the extended storage medium supported by the electronic device. The computer - readable storage medium provides a storage space, and this storage space stores the processing system of the electronic device.

[0058] Moreover, in this storage space, one or more instructions suitable for being loaded and executed by a processor are also stored. These instructions can be one or more computer programs (including program codes). It should be noted that the computer - readable storage medium here can be a high - speed RAM memory or a non - volatile memory, such as at least one disk memory; optionally, it can also be at least one computer - readable storage medium located far from the aforementioned processor.

[0059] In one embodiment, one or more instructions are stored in the computer - readable storage medium; the processor loads and executes one or more instructions stored in the computer - readable storage medium to implement the following process: Based on the method for constructing an optimal locally repairable code of an arc in a projective plane described in Example 1, a parity - check matrix of the locally repairable code is given ; Let represent a linear code with as the parity - check matrix. For any codeword in , its coordinate components are divided into groups of every for a total of groups; For , the sub - matrix in the parity - check matrix and any codeword in the linear code of the group coordinate components are multiplied to obtain the corresponding parity - check equation, and according to the parity - check equation, the repair scheme of each failed node in the

[0060] Example 8: This implementation provides a computer program product or a computer program. The computer program product or the computer program includes computer instructions, and the computer instructions are stored in a computer-readable storage medium. A processor of an electronic device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, causing the electronic device to perform the following process: Based on the method for constructing an optimal locally repairable code of an arc in a projective plane described in Example 1, a parity-check matrix of the locally repairable code is given ; Let denote a linear code with as the parity-check matrix. For any codeword in , its coordinate components are grouped into groups of every components, and a total of groups are obtained; For , the product of the th submatrix in the parity-check matrix and the th group of coordinate components of any codeword in the linear code gives a corresponding parity-check equation. According to the parity-check equation, a repair scheme for each failed node in the th group is obtained.

[0061] Those of ordinary skill in the art can realize that the units and algorithm steps of the examples described in combination with the embodiments disclosed in this application can be implemented by electronic hardware, or by a combination of computer software and electronic hardware. Whether these functions are executed in a hardware or software manner depends on the specific application and design constraints of the technical solution. Those of ordinary skill in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered to exceed the scope of this application.

[0062] ​In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the processes or functions according to the embodiments of the present application are generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer instructions can be stored in a computer-readable storage medium or transmitted through a computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center in a wired manner (e.g., coaxial cable, optical fiber, digital subscriber line (DSL)) or a wireless manner (e.g., infrared, wireless, microwave, etc.). The computer-readable storage medium can be any available medium that can be accessed by a computer or a data processing device such as a server or a data center that includes one or more integrated available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid state disk (SSD)), etc.

[0063] The foregoing is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A construction method of an optimal locally repairable code based on arcs in the projective plane, characterized in that, Including the following processes: Let denote the maximum number of faulty nodes that can be tolerated during the local repair process; let be a prime power and satisfy ; let be a positive integer that satisfies , and when is odd,[[]] ; when is even,[[]] ; When is odd, let the point set be an oval in the projective plane ; when is even, let the point set be a hyperoval in the projective plane ; take points from the point set , and divide every points into a group to form a point set, obtaining point sets; Determine each parameter of the parity-check matrix according to the properties of the points in each point set. For , the -th sub-matrix of the local repair group of the parity-check matrix exactly forms a standard generator matrix of an MDS code with parameters , having a locality of , a minimum distance of , and generating an optimal locally repairable code with disjoint repair groups and parameters .

2. The method for constructing an optimal locally repairable code based on an arc in a projective plane according to claim 1, characterized in that: The parity-check matrix H is: ； Among them, is the identity matrix of order denotes the all-zero matrix in ; for each is a column vector in and each is a column vector in ; in the case of an optimal locally repairable code with disjoint repair groups and the number of repair groups , .

3. The method for constructing an optimal locally repairable code based on an arc in a projective plane according to claim 2, characterized in that: Take out from the point set and take out points, and divide every points into a group to form a point set, obtaining point sets, denoted as , and each point set is expressed as: ; ; …; ; For any , the point set satisfies: , and any three points in the set are not collinear.

4. The method for constructing an optimal locally repairable code based on an arc in a projective plane according to claim 2, characterized in that: According to the obtained point set , for , , calculate the dual subspace of each point in , let , and obtain line sets ; For , each set of lines has lines, , for any , the set of lines satisfies: , and any three lines in the set do not intersect at the same point; For , take three lines in , let , , , where , and respectively represent the one-dimensional spaces spanned by the non-zero vectors in , , , which are three points in the projective plane . The obtained vectors , and are used as the lower half vectors of the th repair group in the parity-check matrix ; For , let and .

5. The method for constructing an optimal locally repairable code based on an arc in a projective plane according to claim 4, characterized in that: Since any three lines do not intersect at the same point, so are all non-zero elements in, and then the upper half of the parameters in the parity-check matrix are: ， ， ； Among them, .

6. The method for constructing an optimal locally repairable code based on an arc in a projective plane according to any one of claims 1-5, characterized in that: When is odd, ; When is even, ; Among them, is a finite field containing elements.

7. A method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane, including the following processes: Construct the parity-check matrix of the local repair code by using the optimal local repair code construction method of arcs in the projective plane according to any one of claims 1-6 ; Let denote the linear code with as the parity-check matrix. For any codeword in , divide its coordinate components into groups of every components, and a total of groups are obtained; For , the product of the -th submatrix in the parity-check matrix and any codeword in the linear code at the -th group of coordinate components gives the corresponding parity-check equation, and based on the parity-check equation, the repair scheme for each failed node in the -th group is obtained. ​ 8. A computer device, characterized in that, Including: A processor and a computer-readable storage medium; The processor is adapted to execute a computer program; The computer-readable storage medium stores a computer program, and when the computer program is executed by the processor, the method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane according to claim 7 is implemented.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program, and the computer program is adapted to be loaded and executed by the processor to implement the method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane according to claim 7.

10. A computer program product, characterized in that, The computer program product includes a computer program, and when the computer program is executed by the processor, the method for repairing a failed node of an optimal locally repairable code based on an arc in a projective plane according to claim 7 is implemented.

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