CN110224790B - Subspace code division greedy method based on Echelon-Ferrs - Google Patents
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Abstract
The invention discloses a subspace code division greedy method based on Echelon-Ferrerers, which comprises the following steps: step one, determining a plurality of given parameters to generate a binary vector set of all combination numbers, and calculating the dimensionality of the binary vector; step two, sorting according to the dimension from large to large, and taking the maximum dimension vector as a first result set; calculating from the maximum dimension smaller than the maximum dimension vector until the dimension is equal to 0, obtaining a second result set of the dimension, and obtaining a third result set of the second result set and the first result set which are compatible; and step four, obtaining all maximum cliques through a maximum complete subgraph algorithm, obtaining the best maximum cliques, adding the best maximum cliques into the first result set, and obtaining a final fourth result set.
Description
Technical Field
The invention relates to the field of network coding error correcting codes, in particular to a subspace code division greedy method based on Echelon-Ferrers.
Background
The current social communication field develops at a high speed, the transmission requirement on a communication network is higher and higher, and the traditional transmission mode cannot meet the increasing demand, so that new theories and technologies are continuously developed. The traditional way of transmitting data in a communication network is storage forwarding, and except for a sending node and a receiving node, an intermediate node in the network is only responsible for routing, that is, only plays a role of forwarding without any processing on the data. On a network
Before coding appears, people generally think that no benefit can be obtained when data of an intermediate node is processed, and the network coding theory proposed by R Ahlswede et al thoroughly overturns the traditional view. Network coding (Network coding) is an information exchange technology combining routing and coding, and the core idea is to allow a Network intermediate node to perform linear or nonlinear processing on data and then forward the data to a downstream node, wherein the intermediate node plays the role of an encoder. Due to the constraint of the maximum flow and the minimum cut theorem in the graph theory, the maximum rate of the communication network must not exceed the maximum flow value between the sending end and the receiving end, and the maximum flow value cannot be reached by the traditional multicast routing method. R Ahlswede et al take the research of butterfly network as an example, and indicate that the maximum stream bound of multicast routing transmission is possible to achieve in network coding, which shows that the network coding can greatly improve the transmission efficiency of information, thereby laying an important position in the communication field.
The network coding has the great significance that: it indicates that the network information stream can be compressed, thereby overriding the classical conclusion that individual bits can no longer be compressed. Network coding allows the intermediate nodes to play the role of signal processors, which, although increasing the complexity of the network, greatly increases the network capacity and the transmission efficiency of information. The network coding has the advantages that: (1) the throughput of the network is improved; (2) improving network load balance; (3) the bandwidth utilization rate is improved; (4) energy consumption of the wireless network nodes is saved. In linear stochastic network coding, Koetter and Kschischang propose linear operator channel models, and subspace coding is increasingly gaining attention. Subspace coding (Subspace coding) is an important component of network coding, and unlike conventional error correction codes, each codeword in Subspace coding is a Subspace, and it is a Subspace distance that is used to measure error detection and correction capabilities of codes. Subspace coding is an important linear network coding.
Koetter et al gives the error detection and correction capabilities of subspace coding: if the subspace distance d satisfies the condition d >2t +2p, the subspace code can correct t packet errors and p erasure errors. Therefore, when the subspace distance d is determined, if the subspace meeting the distance condition can be obtained as many as possible in the limited space, more code words can be obtained, and the information transmission efficiency is improved. How to obtain more code numbers in a limited space is an important research topic in the field of subspace coding.
Disclosure of Invention
The invention designs and develops a subspace code division greedy method based on Echelon-Ferrers, and aims to solve the problem of acquiring more code numbers in a limited space.
The technical scheme provided by the invention is as follows:
a subspace code division greedy method based on Echelon-Ferrers comprises the following steps:
step one, determining a plurality of given parameters to generate a binary vector set of all combination numbers, and calculating the dimensionality of the binary vector;
step two, sorting according to the dimension from large to large, and taking the maximum dimension vector as a first result set;
calculating from the maximum dimension smaller than the maximum dimension vector until the dimension is equal to 0, obtaining a second result set of the dimension, and obtaining a third result set of the second result set and the first result set which are compatible;
and step four, obtaining all maximum cliques through a maximum complete subgraph algorithm, obtaining the best maximum cliques, adding the best maximum cliques into the first result set, and obtaining a final fourth result set.
Preferably, in the first step, the process of calculating the dimensionality of the binary vector comprises:
and expressing the vector according to a simplest row-column ladder matrix, and determining that the number of rows and columns in the matrix is removed according to the given parameters, wherein the remaining minimum number of points is the dimension.
Preferably, in the fourth step, the maximal complete subgraph algorithm includes:
firstly, determining a set C, a graph T and a set D of the maximum complete subgraph;
then, by calling
Starting depth-first traversal, and when the set C and the graph T are both empty sets, representing a maximum complete sub-graph generation;
finally, selecting a vertex Vp with the maximum degree from the union set of the T & ltU & gt D every time, traversing each vertex v in the set T', adding v into the candidate set C, recursively calling an MCE algorithm once for each vertex, using whether other vertices can generate a maximum complete subgraph, removing the vertex v, and then putting back the vertex v into the set D again;
where N (Vp) is the set of all points adjacent to Vp, and T' is the set of vertex sets N (Vp) connected to Vp subtracted from the set T.
Preferably, in the third step, the hamming distance of the vectors consistent with the first result set and the second result set is not less than the given parameter.
Preferably, in the fourth step, the best maximum cluster is the maximum cluster with the largest number of codewords.
Preferably, the simplest row-column ladder matrix is:
the initial coefficient of each row is positioned at the right side of the initial coefficient of the previous row; all leader coefficients are 1; all coefficients located to the left of the leader coefficient are 0; each item leader coefficient is the only nonzero element positioned in the column;
wherein the leader coefficient is the first element in a row that is not 0.
Compared with the prior art, the invention has the following beneficial effects: the method provided by the invention can improve the number of the existing subspace code words, can obtain the subspace meeting the distance condition as much as possible in the limited space, can also obtain more code words, and improves the information transmission efficiency.
Drawings
Fig. 1 is 2 completely large subgraphs as described in example 1 of the present invention.
Detailed Description
The present invention is further described in detail below with reference to the attached drawings so that those skilled in the art can implement the invention by referring to the description text.
The invention provides a subspace code division greedy method based on Echelon-Ferrerers, which comprises the following steps:
step one, given parameters n, k and d, generating all combination numbers
A binary vector set with the length of n and the weight of k is recorded as iVset;
step two, calculating the dimensionality of the vectors;
step three, sorting according to the dimension from large to small;
step four, putting the max vector of the maximum dimension into a result set Sv;
step five, calculating from the second largest dimension until the dimension is equal to 0; namely, the dimension is equal to max-1 from i, 1 is reduced each time until 0, all vector sets with the dimension being i are obtained, and all sets iSet which are compatible with the result set Sv are found; calling an existing maximum complete subgraph algorithm MCE (phi, iSet, phi) to generate all maximum cliques; selecting the best maximum cluster from the result sets and adding the best maximum cluster into the result set sv;
and step six, returning the result set Sv, namely the final result set.
In another embodiment, the dimension is calculated by firstly representing the vector according to a simplest row-column ladder matrix; wherein, the simplest row-column ladder matrix is defined as follows:
1) the leading coefficient of each row (the first element in a row that is not 0) is located to the right of the leading coefficient of the previous row;
2) all leader coefficients are 1;
3) all the coefficients located at the left side of the leader coefficient are 0
4) Each item leader coefficient is the only nonzero element positioned in the column;
in this embodiment, preferably, when n is 6, k is 3, and d is 4, the vector (110100) is expressed as follows according to the simplest row-column ladder matrix:
Setting parameters
After removing the i rows and the delta-i columns from the above figure, the minimum number of points remaining is the dimension, which is therefore 5 in this embodiment, depending on the calculation process.
In another embodiment, the maximum complete subgraph algorithm MCE needs to input three parameters: set C, graph T, set D:
{
The output set C is a very large complete subgraph;
the program is ended;
}
1) removing vertex v from T; recursively calling MCE (union of set C and { v }, set T and N (v) and set D and N (v));
2) adding the vertex v into the set D;
the further explanation of the steps comprises the following steps: all the maximal complete subgraphs of the graph T returned by the MCE algorithm are generated by a method of traversing the whole graph deeply by a backtracking method, and the algorithm calls
Starting depth-first traversal, and when C and T are both empty sets, representing a maximum complete sub-graph generation; selecting a vertex Vp with the maximum degree from the T U D union each time, traversing each vertex v in T', adding v into the candidate set C, recursively calling an MCE algorithm for each vertex, and finally backtracking to try to use other vertices to generate a maximum complete subgraph or not and remove the vertex v, so that the vertex V is replaced and the D is removed again; where N (Vp) represents the set of all points adjacent to Vp.
Example 1
As shown in fig. 1, running the above algorithm results in 2 completely maximal sub-graphs { {0,2,3}, {0,1,3,4} }, the implementation process of the algorithm:
three parameters are input: collection
Graph T {0,1,2,3,4}, set
At the time of starting the operation of the device,
step one, the set T is not an empty set and does not meet the condition;
secondly, selecting a vertex Vp (0) with the maximum degree from the union set T and D;
third, T' is equal to the set T ═ {0,1,2,3,4} minus {1,2,3,4}, T ═ 0 };
step four, v ═ 0 belongs to the set T', and the following two steps are performed:
1) t ═ 1,2,3,4, call
2)D＝{0}；
Maximal complete subgraph algorithm
When the machine is in operation,
step one, the set T is not an empty set and does not meet the condition;
secondly, selecting a vertex Vp ═ 3} with the maximum degree from the union set T ═ D;
third, T 'is equal to the set T ═ {1,2,3,4} minus {1,2,4}, T' ═ 3 };
step four, v ═ 3 belongs to the set T', and the following two steps are performed:
1) t ═ 1,2,4, call
2) D is added to the vertex v, and D is {3 };
maximal complete subgraph algorithm
When the machine is in operation,
step one, the set T is not an empty set and does not meet the condition;
secondly, selecting a vertex Vp (1) with the maximum degree from the union set T and D;
third, T 'is equal to the set T ═ {1,2,4} minus {2,4}, T' ═ 1 };
step four, v ═ 1 belongs to the set T', and the following two steps are performed:
1) t ═ 2,4, call
2) D is added to the vertex v, and D is {1 };
maximal complete subgraph algorithm
When the machine is in operation,
step one, the set T is not an empty set and does not meet the condition;
secondly, selecting a vertex Vp (4) with the maximum degree from the union set T and D;
third, T 'is equal to the set T {2,4} minus {2}, T' {4 };
step four, v ═ 4 belongs to the set T', and the following two steps are performed:
1) t ═ 2, call
2) D is added to the vertex v, and D is {1 };
maximal complete subgraph algorithm
When the machine is in operation,
in the first step, the condition is satisfied and {0,1,3,4} is output.
In another embodiment, the best maximum clique is the one with the highest number of codewords.
Example 2
When n is 6, d is 4, k is 3, q is generally 2,3,4,5,7,8,9, all of which are in common
And vectors, wherein the Hamming distance of any two vectors in a subset S is to be found out, the Hamming distance of any two vectors in the subset S is more than or equal to 4, and the sum of the number (the dimension power of q) of the corresponding code words of the vectors in the subset S is as much as possible.
Step one, generating all 20 vectors as shown in table 1
TABLE 1
Step two, defining the distance d to be 4 according to the dimensionality in the method of the patent, wherein the computed dimensionality is shown in the table above;
step three, sorting the components according to the dimension in the table:
v1，v2，v3，v5，v4，v6，v11，v7，v8，v12，v13，v9，v14，v15，v10，v16，v17，v18，v19，v20
the fourth step of mixing v1Put result set Sv ═ { v ═ v1}
The fifth step, according to v1Dimension i is calculated from 5 up to 0;
when i is 5, 4, 3, according to hamming distance, no addition is made because of incompatibility with the result set;
when i is 2, there is v according to Hamming distance8Can be added;
when i is 1, there is v according to Hamming distance15Can be added;
when i is 0, there is v according to Hamming distance19Can be added;
sixthly, the result set Sv is set as { v ═ v1，v8，v15，v19Returning;
the algorithm is applied to obtain 4 vectors { v } in S1，v8，v15，v19According to the construction method, the lower bound of the code word of Aq (6, 4, 3) exceeds q6+q2+q+1。
Example 3
As shown in table 2, n is 13, k is 5, d is 4, and q is prime power, and is generally 2,3,4,5,7,8, and 9, which are common
And vectors, wherein the Hamming distance of any two vectors in a subset S is to be found out, the Hamming distance of any two vectors in the subset S is more than or equal to 4, and the sum of the number (the dimension power of q) of the corresponding code words of the vectors in the subset S is as much as possible. Applying the above algorithm results in 100 vectors in S, 24 of which are in the following table, according to the construction method according to the above algorithm, the code word lower bound of Aq (13, 4, 5) exceeds q32+q28+q26+8q24+q23+3q22+q21+4q20+4q19+4q18+4q17+4q16+6q15+12q14+7q13+6q12+5q11+2q10+8q9+4q8+3q7+q6+4q4+q3+3q2+q+1
TABLE 2
| (Vector) | Dimension number | (Vector) | |
||
| 1 | 1111100000000 | 32 | 13 | 0110101010000 | 22 |
| 2 | 1110011000000 | 28 | 14 | 0110110001000 | 22 |
| 3 | 1101010100000 | 26 | 15 | 0111001001000 | 22 |
| 4 | 1011001100000 | 24 | 16 | 1010101001000 | 21 |
| 5 | 1001111000000 | 24 | 17 | 1110000001100 | 20 |
| 6 | 1100110010000 | 24 | 18 | 0101110000100 | 20 |
| 7 | 1010110100000 | 24 | 19 | 0101100110000 | 20 |
| 8 | 1110000110000 | 24 | 20 | 0111000100100 | 20 |
| 9 | 1100101100000 | 24 | 21 | 0011100101000 | 19 |
| 10 | 0111010010000 | 24 | 22 | 0011101000100 | 19 |
| 11 | 1101001010000 | 24 | 23 | 0011110000010 | 19 |
| 12 | 1011010001000 | 23 | 24 | 1011000010100 | 19 |
Based on the prior art, the greedy _ multicomponent lower bound on the Aq (13, 4, 5) codeword is currently as follows:
q32+q28+q26+8q24+q23+q22+3q21+2q20+3q19+5q18+5q17+9q16+q15+3q14+3q13+3q12+6q11+3q10+q9+8q8+q7+3q6+3q5+4q4+4q3+2q2+q+1
the lower bound is not as large as the number of code words in the method of the invention, and the calculation finds that there are 161 lower bound improvements, including the following results:
aq (13, 4, 5), Aq (14, 4, 5), Aq (13, 4, 6), Aq (14, 4, 6), Aq (15, 4, 6), Aq (16, 4, 6), Aq (17, 4, 6), Aq (15, 4, 7), Aq (16, 4, 7), Aq (17, 4, 7), Aq (18, 4, 7), Aq (19, 4, 7), Aq (15, 6, 6), Aq (16, 6, 6), A q (17, 6, 6), Aq (18, 6, 6), Aq (19, 6, 6), Aq (14, 6, 7), Aq (15, 6, 7), Aq (16, 6, 7), Aq (17, 6, 7), Aq (18, 6, 7), Aq (19, 6, 7), wherein {2, 3,4,5, 8,9, 8 }, may be used as {2, 3,4, 7 }, or {2, 6, 7 }, or 4.
From the increasing ratio, the code word which is increased most is A2(19, 4, 7), the code word is increased by 11.5% on the basis of the original code word, the minimum code word is A2(13, 4, 6), the number of the code words is increased from 38325127529 to 38325131657, and the code word is increased by 1.07 multiplied by 10-7。
While embodiments of the invention have been described above, it is not limited to the applications set forth in the description and the embodiments, which are fully applicable in various fields of endeavor to which the invention pertains, and further modifications may readily be made by those skilled in the art, it being understood that the invention is not limited to the details shown and described herein without departing from the general concept defined by the appended claims and their equivalents.
Claims (5)
1. A subspace code division greedy method based on Echelon-Ferrs is characterized by comprising the following steps:
step one, determining a plurality of given parameters to generate a binary vector set of all combination numbers, and calculating the dimensionality of the binary vector;
step two, sorting according to the dimension from large to small, and taking the maximum dimension vector as a first result set;
calculating from the maximum dimension smaller than the maximum dimension vector until the dimension is equal to 0, obtaining a second result set of the dimension, and obtaining a third result set of the second result set and the first result set which are compatible;
step four, obtaining all maximum cliques through a maximum complete subgraph algorithm, obtaining the best maximum cliques, adding the best maximum cliques into the first result set, and obtaining a final fourth result set;
wherein the maximal complete subgraph algorithm comprises:
firstly, determining a set C, a set T and a set D of the maximum complete subgraph;
then, by calling
Starting depth-first traversal, and when the set C and the set T are both empty sets, representing a maximum complete sub-graph generation;
finally, selecting a vertex Vp with the maximum degree from the union set of the T & ltU & gt D every time, traversing each vertex v in the set T', adding v into the candidate set C, recursively calling an MCE algorithm once for each vertex, using other vertices to generate a maximum complete subgraph, removing the vertex v from the set T, and then putting back the vertex V in the set D;
where N (Vp) is the set of all points adjacent to Vp, and T' is the set of vertex sets N (Vp) connected to Vp subtracted from the set T.
2. The Echelon-Ferrers-based subspace code division greedy method of claim 1, wherein in the step one, the process of calculating the dimension of the binary vector comprises:
and expressing the vector according to a simplest row-column ladder matrix, and determining that the number of rows and columns in the matrix is removed according to the given parameters, wherein the remaining minimum number of points is the dimension.
3. The Echelon-Ferrers-based subspace code-division greedy method of claim 1, wherein at the third step, the compatibility is such that the hamming distance of the vectors of the first result set and the second result set is not less than the given parameter.
4. The Echelon-Ferrers-based subspace code division greedy method of claim 1, wherein in the fourth step, the best maximum clique is the maximum clique with the largest number of codewords.
5. The Echelon-Ferrers-based subspace code division greedy method of claim 2, wherein the simplest column-rank ladder matrix is:
the initial coefficient of each row is positioned at the right side of the initial coefficient of the previous row; all leader coefficients are 1; all coefficients located to the left of the leader coefficient are 0; each item leader coefficient is the only nonzero element positioned in the column;
wherein the leader coefficient is the first element in a row that is not 0.
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