Greedy selection method for identification vector of multi-construction constant dimension code

Abstract

A greedy selection method for an identification vector of a multi-construction constant dimension code relates to the field of network coding error correction codes, and comprises the following steps: step 1, determining a plurality of given parameters, and generating all binary vector sets meeting conditions; step 2, calculating dimensions of the binary vector; step 3, sorting according to the dimensions from small to large; step 4, starting calculation from the maximum dimension until the dimension is equal to 0, randomly selecting a maximum dimension vector, and if the result set is empty or compatible with the result set, adding the maximum dimension vector into the result set; 5, according to given parameters, calculating the number of code words of multiple construction subspace codes based on the result set vector; and step 6, repeating the step 3 and the step 4 for iter times, and returning a result set with the maximum number of constructed code words. Compared with the prior art, the method provided by the invention can improve the number of the existing constant dimension code words and improve the information propagation efficiency.

Description

Technical Field

[0001] The present invention relates to the field of network coding error correction codes, and in particular to a greedy selection method for identification vectors of multiple constructed constant-dimensional codes. Background Art

[0002] Network coding is an emerging technology that can effectively improve transmission efficiency, reduce latency and enhance network robustness by intelligently encoding and decoding data in the network.

[0003] In 2000, Ahlswede et al. proposed the network coding theory, which pioneered the idea that data can be transmitted in the network through coding, thereby improving the network's capacity and reliability. Unlike traditional routing methods, network coding allows nodes to encode the received data before forwarding it, thereby achieving more efficient data transmission. The advantages of network coding are: (1) improving network throughput; (2) improving network load balancing; (3) improving bandwidth utilization; and (4) saving energy consumption of wireless network nodes.

[0004] Subspace codes were first proposed by Koetter and Kschischang in 2008 to solve the error control problem in random network coding. The key idea of ​​subspace codes is to encode and decode subspaces (rather than specific vectors) so that even if errors or packet loss occur during network transmission, the receiving node can still recover the correct data from the received subspace combination. This coding method can not only effectively combat random errors and packet loss problems in the network, but also improve the transmission efficiency and reliability of the network.

[0005] As an important branch of subspace codes, constant-dimensional codes play an important role in random network coding due to their efficient encoding and decoding algorithms. Koetter et al. gave the error detection and correction capabilities of subspace coding: if the subspace distance d satisfies the condition d>2t+2p, then this subspace code can correct t packet errors and p erasure errors. Therefore, when the subspace distance d is determined, if as many subspaces as possible that meet the distance condition can be obtained in a limited space, more codewords can be obtained, thereby improving the efficiency of information transmission. How to obtain more codewords in a limited space is an important research topic in the field of subspace coding. Based on this, the present invention designs a greedy selection method for identification vectors of multiple constructed constant-dimensional codes. Summary of the invention

[0006] The number of codewords constructed by linking is The number of code words of multiple constructions is the sum of the power of the dimension of q. Combining these two constructions, the present invention designs and develops a greedy selection method for identification vectors of multiple construction constant-dimensional codes. The purpose of the present invention is to solve the problem of obtaining more code words in a limited space.

[0007] A greedy selection method for identification vectors of multiple constructed constant-dimensional codes comprises the following steps:

[0008] Step 1: Determine multiple given parameters and generate a set of all binary vectors that meet the conditions;

[0009] Step 2: Calculate the dimension of the binary vector;

[0010] Step 3: Sort the dimensions from small to large, and add the vector with the largest dimension to the result set;

[0011] Step 4: Start calculating from the maximum dimension until the dimension is equal to 0, randomly select the maximum dimension vector, and add it to the result set if the result set is empty or compatible with the result set;

[0012] Step 5: Calculate the number of codewords of the multi-construction subspace code based on the result set vector according to the given parameters;

[0013] Step 6: Repeat steps 3 and 4 iter times and return the result set with the largest number of constructed codewords.

[0014] Preferably, in step 1, the given parameters include: q, n 1 ,n 2 ,d,k,iter.

[0015] Preferably, in step 1, the set of binary vectors satisfying the condition refers to: a set of binary vectors having a length of n 1 +n 2 , a vector with weight k, and n 2 The number of 1s in each position is at least d / 2.

[0016] Preferably, in step 2, the process of calculating the dimension of the binary vector includes: representing the vector according to a simplest row-column echelon matrix, and determining, based on the given parameters, that the minimum number of points remaining after removing the number of rows and columns in the matrix is ​​the dimension.

[0017] Preferably, in the step 4, the compatibility is that the Hamming distance between the randomly selected maximum dimensional vector and the vectors in the result set is not less than the given parameter d.

[0018] Preferably, the simplest row-column echelon matrix is: the leading coefficient of each row is located to the right of the leading coefficient of the previous row; all leading coefficients are 1; all the leading coefficients to the left are 0; each leading coefficient is the only non-zero element in the column; wherein the leading coefficient is the first non-zero element in a row,

[0019] Preferably, in step five, the process of calculating the number of codewords of the multi-constructed subspace code based on the result set vector is: first calculate the dimension of the vector in the result set, and then calculate the sum of the powers of the dimension of q.

[0020] Compared with the prior art, the method proposed by the present invention can improve the number of existing constant-dimensional code words and improve the efficiency of information transmission. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is the complete maximal subgraph in Embodiment 2 of the present invention. DETAILED DESCRIPTION

[0022] The present invention is described in further detail below so that those skilled in the art can implement it according to the description.

[0023] The present invention provides a greedy selection method for identification vectors of multiple constructed constant-dimensional codes, comprising the following steps:

[0024] Step 1: Given parameter n 1 ,n 2 ,d,k,iter,generate all the length n, weight k and n 2 The set of binary vectors whose number of position 1s is at least d / 2 is called the candidate element set, denoted by V n ;

[0025] Step 2: Calculate the dimensions of these vectors;

[0026] Step 3: Sort by dimension from small to large;

[0027] Step 4: Start from the maximum dimension m, and decrease by 2 each time until the dimension m is equal to 0. n Randomly select a vector with a dimension greater than m-2, and add it to the result set V if the result set is empty or compatible with the result set S ;

[0028] Step 5: Calculate the number of codewords of the multi-construction subspace code based on the result set vector, that is, the sum of the power of the dimension of q;

[0029] Step 6: Repeat steps 4 and 5 iter times and return the result set with the largest number of constructed codewords.

[0030] Example 1

[0031] To calculate the dimension, we first need to express the vector in terms of the simplest row-column echelon matrix. The simplest row-column echelon matrix is ​​defined as follows:

[0032] 1) The leading coefficient of each row (the first non-zero element in a row) is located to the right of the leading coefficient of the previous row;

[0033] 2) All leading coefficients are 1;

[0034] 3) All the coefficients on the left side of the first term are 0;

[0035] 4) The first coefficient of each term is the only non-zero element in the corresponding column;

[0036] In this embodiment, as a preferred embodiment, assuming n=6, k=3, d=4, the vector v=(101010) is expressed as follows according to the simplest row-column ladder matrix: Converted to ferrers figure (Remove 0 and 1 and align to the right).

[0037] Set parameters After removing the common i rows and δ-i columns from the above figure, the minimum number of points remaining is the dimension. According to the calculation process, the dimension in this embodiment is 3, recorded as dim(v)=3, and the number of multi-constructed codewords corresponding to the vector is q 3

[0038] Example 2

[0039] Assume n 1 =3,n 2 =4, d=4, k=4, iter=1, q is usually 2, 3, 4, 5, 7, 8, 9. We need to find a subset V from the set of binary vectors with a length of 7, a weight of 4, and the number of 1s in the last 4 positions is at least 2. S , V S The Hamming distance between any two vectors in V is at least 4, and V S The sum of the number of codewords corresponding to the vectors in (the power of the dimension of q) is as large as possible.

[0040] The first step is to generate all binary vectors with a length of 7, a weight of 3, and the number of 1s in the last 4 positions is at least 2, a total of 22 vectors as shown in Table 1.

[0041] Table 1

[0042] vector Dimensionality vector Dimensionality <![CDATA[v 1 =0000111]]> 0 <![CDATA[v 12 =0011001]]> 2 <![CDATA[v 2 =0001011]]> 0 <![CDATA[v 13 =0100110]]> 2 <![CDATA[v 3 =0001101]]> 0 <![CDATA[v 14 =0101001]]> 2 <![CDATA[v 4 =0001110]]> 0 <![CDATA[v 15 =1000110]]> 2 <![CDATA[v 5 =0010011]]> 0 <![CDATA[v 16 =1001001]]> 2 <![CDATA[v 6 =0100011]]> 0 <![CDATA[v 17 =0011010]]> 2 <![CDATA[v 7 =1000011]]> 0 <![CDATA[v 18 =1001010]]> 3 <![CDATA[v 8 =0010110]]> 1 <![CDATA[v 19 =0101010]]> 3 <![CDATA[v 9 =1000101]]> 1 <![CDATA[v 20 =0011100]]> 3 <![CDATA[v 10 =0100101]]> 1 <![CDATA[v 21 =0101100]]> 4 <![CDATA[v 11 =0010101]]> 1 <![CDATA[v 22 =1001100]]> 4

[0043] Step 2: Distance d = 4. According to the definition of dimension in the method of this patent, the calculated dimension is shown in the table above.

[0044] Step 3: Sort by dimension from small to large, as shown in the table above

[0045] Step 4: Calculate the dimension m from 4 to 0;

[0046] When m=4, we can choose a vector with dimension equal to 3 or 4 and randomly select v 22 , because the result set is empty, it can be added; randomly select v 19 , because it is compatible with the result set, it can be added; because other vectors are incompatible with the result set, they cannot be added;

[0047] When m=2, we can choose a vector with dimension equal to 1 or 2, and randomly select v 12 , because it is compatible with the result set, it can be added; other vectors are incompatible with the result set and cannot be added;

[0048] When m=0, we can select a vector with dimension equal to 0 and randomly select v 1 , because it is compatible with the result set, it can be added; because other vectors are incompatible with the result set, they cannot be added;

[0049] The fifth step is to calculate the number of codewords of the multi-construction subspace code based on the result set vector, that is, the sum of the power of the dimension of q, and apply the above algorithm to obtain V S There are 4 vectors {v 22 ,v 19 ,v 12 ,v 1}, the corresponding number of codewords is q 4 +q 3 +q 2 +1.

[0050] Step 6: Repeat steps 4 and 5. Since iter = 1, it only needs to be repeated once. The dimension m is calculated from 4 to 0.

[0051] When m=4, we can choose a vector with dimension equal to 3 or 4 and randomly select v 21 , because the result set is empty, it can be added; randomly select v 18 , because it is compatible with the result set, it can be added; because other vectors are incompatible with the result set, they cannot be added;

[0052] When m=2, we can choose a vector with dimension equal to 1 or 2, and randomly select v 9 , because it is compatible with the result set, it can be added;

[0053] Randomly select v 13 , because it is compatible with the result set, it can be added; randomly select v 8 , because it is compatible with the result set, it can be added; other vectors are incompatible with the result set and cannot be added;

[0054] When m=0, we can select a vector with dimension equal to 0; randomly select v 6, because it is compatible with the result set, it can be added;

[0055] Because the other vectors are incompatible with the result set, they cannot be added;

[0056] Calculate the number of codewords of the multi-construction subspace code based on the result set vector, that is, the sum of the power of the dimension of q. Applying the above algorithm, we get:

[0057] V S = {v 21 ,v 18 ,v 13 ,v 9 ,v 8 ,v 6}

[0058] The corresponding number of code words is:

[0059] q 4 +q 3 +q 2 +2q+1.

[0060] Finally, because:

[0061] q 4 +q 3 +q 2 +2q+1>q 4 +q 3 +q 2 +1

[0062] So the result set with the largest number of codewords is returned:

[0063] V S = {v 21 ,v 18 ,v 13 ,v 9 ,v 8 ,v 6}

[0064] The graph formed by the elements of this set is a complete maximal subgraph of the graph formed by the candidate elements.

[0065] Example 3

[0066] Assume n 1 =8,n 2 =6, k=4, d=4, q is a prime power, usually 2, 3, 4, 5, 7, 8, 9, length is 14, weight is 4, and the number of 1s in the last 6 positions is at least 2. There are 595 binary vectors in total. We need to find a subset V from them. S , V S The Hamming distance between any two vectors in V is at least 4, and V SThe sum of the number of codewords (the power of the dimension of q) corresponding to the vectors in is as large as possible.

[0067] Using the above algorithm, we can get V S There are 60 vectors in , as shown in Table 2. According to the multiple construction method, V S The corresponding number of codewords is

[0068] q 18 +2q 16 +3q 15 +5q 14 +6q 12 +7q 11 +9q 10 +7q 9 +8q 8 +5q 7 +3q 6 +q 4 +q 3 +q 2 +1;

[0069] Combined with the link construction, we can get

[0070] A q (14,4,4)≥A q (8,4,4)q 18 +q 18 +2q 16 +3q 15 +5q 14 +6q 12 +7q 11 +9q 10 +7q 9 +8q 8 +5q 7 +3q 6 +q 4 +q 3 +q 2 +1;

[0071] So A 2 (14,4,4)≥1259181405.

[0072] Table 2

[0073] vector Dimensionality vector Dimensionality 11000000110000 18 00000110011000 11 00110000110000 16 00010100100001 10 10100000101000 16 00001001100100 10 00001100110000 14 00101000001001 9 01100000011000 15 10000010100001 9 01100000100100 15 00100010001010 8 01010000101000 15 00011000000110 9 10010000100100 14 10001000000101 8 10100000010100 14 01000100010001 9 10010000011000 14 00110000000011 8 11000000001100 14 10000100001001 8 00110000001100 12 01010000000101 9 10001000100010 12 00010001100010 9 00000011110000 12 00010100001010 9 01010000010010 12 00100010010001 8 00001010101000 12 00000011001100 8 01001000010100 12 00100001100001 8 00010100010100 11 01000100000110 8 00100100100010 11 10000010000110 7 00011000010001 10 01000010001001 7 01001000100001 11 00010001001001 6 00000101101000 11 00100001000110 6 01000010100010 10 00001100000011 6 00001100001100 10 00100100000101 7 00001001011000 10 10000001010001 7 00101000010010 11 10000001001010 7 00000110100100 11 00000000111001 3 01001000001010 10 00000011000011 4 11000000000011 10 00000000110110 2 10000100010010 10 00000000001111 0

[0074] Based on the existing technology, the current 2 The lower bound of the (14,4,4) codeword is 1259181253, which is not as large as the number of codewords in the method of the present invention.

[0075] Example 4

[0076] Assume n1 =12,n 2 =6, k=4, d=4, q is a prime power, usually 2, 3, 4, 5, 7, 8, 9, length is 18, weight is 4, and the number of 1s in the last 6 positions is at least 2. There are 1245 binary vectors in total. We need to find a subset V from them. S , V S The Hamming distance between any two vectors in V is at least 4, and V S The sum of the number of codewords (the power of the dimension of q) corresponding to the vectors in is as large as possible. Apply the above algorithm to get V S There are 84 vectors in , as shown in Table 4. According to the multiple construction method, V S The corresponding number of code words is: q 22 +2q 20 +3q 19 +5q 18 +6q 16 +7q 15 +10q 14 +6q 13 +12q 12 +9q 11 +8q 10 +q 9 +5q 8 +2q 7 +3q 6 +2q 4 +q 2 +1

[0077] Combined with the link construction, we get:

[0078] A q (18,4,4)≥A q (12,4,4)q 18 q 22 +2q 20 +3q 19 +5q 18 +6q 16 +7q 15 +10q 14 +6q 13 +12q 12 +9q 11 +8q 10 +q 9 +5q 8 +2q 7 +3q 6 +2q 4 +q 2 +1

[0079] So A 2 (18,4,4)≥5158164354661.

[0080] Table 3

[0081] vector Dimensions vector Dimensions 110000000000110000 22 010000010000100001 12 101000000000101000 20 010000100000001010 12 001100000000110000 20 010100000000000101 13 010100000000101000 19 001001000000010001 13 100100000000100100 18 000000110000001100 12 110000000000001100 18 001010000000001001 13 100100000000011000 18 000100100000010001 12 101000000000010100 18 001001000000001010 13 011000000000011000 19 000000001010101000 12 011000000000100100 19 000000000011110000 12 000011000000110000 18 100000010000000110 10 100010000000100010 16 001000000100100010 11 000110000000010100 16 000011000000000011 10 001100000000001100 16 000000000110011000 11 000010100000101000 16 001000100000000110 11 010100000000010010 16 100000001000001010 10 000000110000110000 16 100000010000010001 11 000110000000100001 15 000000101000010010 11 000001100000011000 15 000000000101101000 11 100001000000010010 14 000000001001011000 10 110000000000000011 14 000000001100001100 10 000110000000001010 14 000100010000001001 10 010001000000010100 15 100000100000001001 11 010001000000100010 15 000000100100010100 11 100001000000100001 14 000000010100010010 10 000001010000101000 15 000000000110100100 11 010010000000010001 14 000000001001100100 10 000010010000011000 14 001000010000000101 9 000100100000100010 14 000000001010010001 8 000011000000001100 14 000001001000000101 8 001010000000010010 15 000000010010001010 8 000010010000100100 14 000000110000000011 8 000001100000100100 15 000000000011001100 8 000000001100110000 14 000000001010000110 7 000101000000000110 12 000000001100000011 6 001000100000100001 13 000000100010000101 6 000000011000100010 12 000000000101010001 7 010001000000001001 12 000000000101000110 6 100010000000000101 12 000000000000111100 4 001100000000000011 12 000000000011000011 4 010010000000000110 13 000000000000110011 2 000000011000010100 12 000000000000001111 0

[0082] Based on the existing technology, the current 2 The lower bound of the (18,4,4) codeword is 5158164354661, which is not as large as the number of codewords in the method of the present invention.

[0083] In addition, calculations show that the method of the present invention is better than the prior art in other parameters, including the following results:

[0084] A q (13,4,4),A q (14,4,4),A q (15,4,4),A q (17,4,4),A q (18,4,4),A q (19,4,4),A q (13,4,5),A q (14,4,5),A q (17,4,5),A q (18,4,5),A q (19,4,5),

[0085] Where q can be {2, 3, 4, 5, 7, 8, 9}.

[0086] Although the embodiments of the present invention have been disclosed as above, they are not limited to the applications listed in the specification and the implementation modes. They can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily implemented. Therefore, without departing from the general concept defined by the claims and the scope of equivalents, the present invention is not limited to specific details.

Claims

1. A greedy selection method for identification vectors of multiple constructed constant-dimensional codes, characterized in that: The steps include: Step 1: Determine multiple given parameters, the given parameters include: q, n1, n2, d, k, iter, and generate a set of all binary vectors that meet the conditions; Step 2: Calculate the dimension of the binary vector; Step 3: sorting the dimensions from small to large; Step 4: Start calculating from the maximum dimension until the dimension is equal to 0, randomly select the maximum dimension vector, and add it to the result set if the result set is empty or compatible with the result set; Step 5: Calculate the number of codewords of the multi-construction subspace code based on the result set vector according to the given parameters; Step 6: Repeat steps 3 and 4 iter times and return the result set with the largest number of constructed codewords.

2. The greedy selection method for identification vectors of multiple construction subspace codes as claimed in claim 1, characterized in that: In step 1, the binary vector that meets the conditions refers to a binary vector with a length of n1+n2, a weight of k, and a number of 1s in the last n2 positions being at least d / 2.

3. The greedy selection method for identification vectors of multiple construction subspace codes as claimed in claim 1, characterized in that: In step 2, the process of calculating the dimension of the binary vector includes: representing the vector according to the simplest row-column echelon matrix, and determining the dimension as the minimum number of points remaining after removing the number of rows and columns in the matrix according to the given parameters.

4. The greedy selection method for identification vectors of multiple construction subspace codes as claimed in claim 1, characterized in that: In step 4, the compatibility is that the Hamming distance between the randomly selected maximum dimensional vector and the vectors in the result set is not less than the given parameter d.

5. The greedy selection method for identification vectors of multiple construction subspace codes as claimed in claim 2, characterized in that: The simplest row-column echelon matrix is: the leading coefficient of each row is located to the right of the leading coefficient of the previous row; all leading coefficients are 1; all to the left of the leading coefficient are 0; each leading coefficient is the only non-zero element in its column; wherein the leading coefficient is the first non-zero element in a row.

6. The greedy selection method for identification vectors of multiple construction subspace codes as claimed in claim 1, characterized in that In step five, the process of calculating the number of codewords of the multi-constructed subspace code based on the result set vector is: first calculate the dimension of the vector in the result set, and then calculate the sum of the power of the dimension of q.

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