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Linear sum-rank-metric codes

This page lists some tables of the dimensions of the linear sum-rank-metric codes with various lengths and the matrix size n*m. For the constructions of SRM codes, we refer to the following references.

  1. U. MartÍnez-Peňas, Sum-rank BCH codes and cyclic-skew-cyclic codes, IEEE Transactions on Information Theory, vol. 67, no. 8, pp. 51495167, 2021.

  2. H. Chen, New explicit linear sum-rank-metric codes, IEEE Transactions on Information Theory, vol. 69, no. 10, pp. 6303-6313, 2023.

  3. H. Chen, Z. Cheng and Y. Qi, Construction and fast decoding of binary linear sum-rank-metric codes, arXiv:2311.03619, 2023.

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Table 1: Block length t=7, q=2, n=m=2

Minimum sum-rank distance
Reference 2
Reference 1 (Table III)
Singleton-like

2

2*13

2*12

2*13

3

2*10

2*9

2*12

4

2*9

2*6

2*11

5

2*6

2*5

2*10

6

2*5

2*5

2*9

7

2*4

2*2

2*8

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Table 2： Block length t=15, q=2, n=m=2

Minimum sum-rank distance
Reference 2
Reference 1 (Table IV)
Singleton-like

4

2*25

2*20

2*27

5

2*21

2*18

2*26

6

2*20

2*16

2*25

7

2*18

2*14

2*24

8

2*17

2*10

2*23

9

2*13

2*8

2*22

10

2*13

2*8

2*21

11

2*11

2*6

2*20

12

2*10

2*4

2*19

13

2*8

none

2*18

14

2*8

2*2

2*17

15

2*7

none

2*16

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Table 3： Block length t=17, q=2, n=m=2

Minimum sum-rank distance
Reference 2
Singleton-like

4

2*29

2*31

5

2*25

2*30

6

2*24

2*29

7

2*22

2*28

8

2*21

2*27

9

2*20

2*26

10

2*16

2*25

11

2*14

2*24

12

2*14

2*23

13

2*11

2*22

14

2*10

2*21

15

2*9

2*20

16

2*8

2*19

17

2*7

2*18

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Table 4： Block length t=31, q=2, n=m=2

Minimum sum-rank distance
Reference 2
Reference 1 (Table V)
Singleton-like

4

2*55

2*50

2*59

5

2*51

2*47

2*58

6

2*50

2*45

2*57

7

2*46

2*40

2*56

8

2*45

2*35

2*55

9

2*41

none

2*54

10

2*40

2*30

2*53

11

2*38

none

2*52

12

2*38

2*25

2*51

13

2*33

none

2*50

14

2*31

2*20

2*49

15

2*29

none

2*48

16

2*28

none

2*47

17

2*25

none

2*46

18

2*24

2*7

2*45

19

2*22

none

2*44

20

2*21

none

2*43

21

2*19

none

2*42

22

2*19

2*7

2*41

23

2*18

none

2*40

24

2*17

none

2*39

25

2*13

none

2*38

26

2*13

2*2

2*37

27

2*12

none

2*36

28

2*12

none

2*35

29

2*11

none

2*34

30

2*11

2*2

2*33

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Table 5： Block length t=63, q=2, n=m=2

Minimum sum-rank distance
Reference 2
Reference 1 (Table VI)
Singleton-like

4

2*119

2*112

2*123

5

2*114

2*108

2*122

6

2*112

2*106

2*121

7

2*107

2*100

2*120

8

2*106

none

2*119

9

2*101

none

2*118

10

2*99

2*88

2*117

11

2*96

none

2*116

12

2*95

none

2*115

13

2*88

none

2*114

14

2*87

2*70

2*113

15

2*84

none

2*112

16

2*82

none

2*111

17

2*77

none

2*110

18

2*77

none

2*109

19

2*72

none

2*108

20

2*71

none

2*107

21

2*70

none

2*106

22

2*69

2*52

2*105

23

2*67

none

2*104

24

2*66

none

2*103

25

2*58

none

2*102

26

2*58

none

2*101

27

2*56

none

2*100

28

2*55

none

2*99

29

2*51

none

2*98

30

2*50

2*28

2*97

31

2*47

none

2*96

32

2*46

none

2*95

38

2*37

2*16

2*89

46

2*29

2*8

2*81

54

2*20

2*8

2*73

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Table 6： Block length t=127, q=2, n=m=2

Minimum sum-rank distance
Reference 2
Reference 1 (Table VII)
Singleton-like

4

2*246

2*238

2*251

5

2*240

2*233

2*250

6

2*236

2*231

2*249

7

2*231

2*224

2*248

10

2*221

2*210

2*245

14

2*202

2*189

2*241

22

2*172

2*154

2*233

30

2*146

2*112

2*225

38

2*124

2*86

2*217

46

2*106

2*70

2*209

54

2*91

2*42

2*201

62

2*77

2*28

2*193

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Table 7： Block length t=31, q=3, n=m=2

Minimum sum-rank distance
Reference 2
Singleton-like

4

2*57

2*59

5

2*53

2*58

6

2*52

2*57

7

2*49

2*56

8

2*47

2*55

9

2*44

2*54

10

2*42

2*53

11

2*40

2*52

12

2*39

2*51

13

2*36

2*50

14

2*35

2*49

15

2*32

2*48

16

2*31

2*47

17

2*29

2*46

18

2*28

2*45

19

2*25

2*44

20

2*24

2*43

21

2*22

2*42

22

2*21

2*41

23

2*20

2*40

24

2*19

2*39

25

2*18

2*38

26

2*17

2*37

27

2*15

2*36

28

2*14

2*35

29

2*13

2*34

30

2*13

2*33

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Table 8： Block length t=21, q=4, n=m=2

Minimum sum-rank distance
Reference 2
Singleton-like

4

2*37

2*39

5

2*33

2*38

6

2*32

2*37

7

2*30

2*36

8

2*29

2*35

9

2*25

2*34

10

2*24

2*33

hashtag
Table 9： Block length t=12, q=2, n=m=3

Minimum sum-rank distance
Reference 2
Singleton-like

4

3*30

3*33

5

3*27

3*32

6

3*26

3*31

7

3*22

3*30

8

3*21

3*29

9

3*19

3*28

10

3*17

3*27

11

3*15

3*26

12

3*15

3*25

13

3*12

3*24

14

3*12

3*23

15

3*11

3*22

16

3*10

3*21

17

3*9

3*20

18

3*9

3*19

19

3*7

3*18

20

3*7

3*17

21

3*6

3*16

22

3*5

3*15

23

3*5

3*14

24

3*5

3*13

hashtag
Table 10： Block length t=31, q=2, n=m=3

Minimum sum-rank distance
Reference 2
Singleton-like

4

3*89

3*90

5

3*83

3*89

6

3*82

3*88

7

3*77

3*87

8

3*75

3*86

9

3*72

3*85

10

3*70

3*84

11

3*67

3*83

12

3*66

3*82

13

3*61

3*81

14

3*59

3*80

15

3*56

3*79

16

3*54

3*78

17

3*52

3*77

18

3*51

3*76

19

3*48

3*75

20

3*47

3*74

21

3*44

3*73

22

3*41

3*72

23

3*40

3*71

24

3*39

3*70

NextBCH and Goppa-type binary linear sum-rank-metric codes