doi: 10.3934/amc.2022039
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New constant dimension subspace codes from multilevel linkage construction

College of Information Science and Technology/Cyber Security, Jinan University, Guangzhou, 510632, Guangdong Province, China

* Corresponding author: Lao Huimin

Received  September 2021 Revised  March 2022 Early access June 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 62032009), and the Guangdong Major Program of Basic and Applied Basic Research (No. 2019B030302008)

One of the main problems in subspace coding is to determine the maximal possible size of a constant dimension subspace code with given parameters. In this paper, we introduce a family of new codes, named rank-restricted Ferrers diagram rank-metric codes, to give an improved construction of constant dimension subspace codes. Our method constructs many new CDCs with larger sizes than previously best-known codes.

Citation: Huimin Lao, Hao Chen. New constant dimension subspace codes from multilevel linkage construction. Advances in Mathematics of Communications, doi: 10.3934/amc.2022039
References:
[1]

H. ChenX. HeJ. Weng and L. Xu, New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inf. Theory, 66 (2020), 5317-5321.  doi: 10.1109/TIT.2020.2975776.

[2]

A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, Advances in Mathematics of Communications. doi: 10.3934/amc.2021007.

[3]

Ph. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.

[4]

T. EtzionE. GorlaA. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Trans. Inf. Theory, 62 (2016), 1616-1630.  doi: 10.1109/TIT.2016.2522971.

[5]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, 55 (2009), 2909-2919.  doi: 10.1109/TIT.2009.2021376.

[6]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory, 57 (2011), 1165-1173.  doi: 10.1109/TIT.2010.2095232.

[7]

É. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredachi Inf., 21 (1985), 3-16. 

[8]

É. M. Gabidulin and M. Bossert, Codes for network coding, Problemy Peredachi Informatsii, 45 (2009), 54-68; translation in Probl. Inf. Transm., 45 (2009), 343-356 doi: 10.1134/S003294600904005X.

[9]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.

[10]

E. Gorla and A. Ravagnani, Subspace codes from Ferrers diagrams, J. Algebra. Appl, 16 (2017), 1750131, 23 pp. doi: 10.1142/S0219498817501316.

[11]

X. HeY. ChenZ. Zhang and K. Zhou, New construction for constant dimension subspace codes via a composite structure, IEEE Communications Letters, 25 (2021), 1422-1426.  doi: 10.1109/LCOMM.2021.3052734.

[12]

D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inf. Theory, 67 (2021), 705-715.  doi: 10.1109/TIT.2020.3038272.

[13]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, arXiv preprint, arXiv: 1601.02864.

[14]

D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, Coding theory and applications, Lecture Notes in Comput. Sci., Springer, Cham, 10495 (2017), 163-191. doi: 10.1007/978-3-319-66278-7_15.

[15]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.

[16]

S. Kurz, Lifted codes and the multilevel construction for constant dimension codes, arXiv preprint, arXiv: 2004.14241.

[17]

S. Kurz, The interplay of different metrics for the construction of constant dimension codes, arXiv preprint, arXiv: 2109.07128. doi: 10.3934/amc.2021069.

[18]

H. LaoH. Chen and X. Tan, New constant dimension subspace codes from block inserting constructions, Cryptography and Communications, 14 (2022), 87-99.  doi: 10.1007/s12095-021-00524-9.

[19]

F. Li, Construction of constant dimension subspace codes by modifying linkage construction, IEEE Trans. Inf. Theory, 66 (2019), 2760-2764.  doi: 10.1109/TIT.2019.2960343.

[20]

S. LiuY. Chang and T. Feng, Parallel multilevel constructions for constant dimension codes, IEEE Trans. Inf. Theory, 66 (2020), 6884-6897.  doi: 10.1109/TIT.2020.3004315.

[21]

Y. NiuQ. Yue and D. Huang, New constant dimension subspace codes from parallel linkage construction and multilevel construction, Cryptogr. Commun., 14 (2021), 201-214.  doi: 10.1007/s12095-021-00504-z.

[22]

Y. NiuQ. Yue and D. Huang, New constant dimension subspace codes from generalized inserting construction, IEEE Communications Letters, 25 (2020), 1066-1069.  doi: 10.1109/LCOMM.2020.3046042.

[23]

Y. Niu, Q. Yue and D. Huang, Construction of constant dimension codes via improved inserting construction, Applicable Algebra in Engineering, Communication and Computing, 1–18. doi: 10.1007/s00200-021-00537-0.

[24]

N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, ferrers diagrams, and pending blocks, IEEE Trans. Inf. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.

[25]

D. SilvaF. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.

[26]

A. TrautmannJ. Rosenthal and A. Edelmayer, New improvements on the echelon-ferrers construction, Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, 19 (2010), 405-408. 

[27]

L. Xu and H. Chen, New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319.  doi: 10.1109/TIT.2018.2839596.

show all references

References:
[1]

H. ChenX. HeJ. Weng and L. Xu, New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inf. Theory, 66 (2020), 5317-5321.  doi: 10.1109/TIT.2020.2975776.

[2]

A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, Advances in Mathematics of Communications. doi: 10.3934/amc.2021007.

[3]

Ph. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.

[4]

T. EtzionE. GorlaA. Ravagnani and A. Wachter-Zeh, Optimal Ferrers diagram rank-metric codes, IEEE Trans. Inf. Theory, 62 (2016), 1616-1630.  doi: 10.1109/TIT.2016.2522971.

[5]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inf. Theory, 55 (2009), 2909-2919.  doi: 10.1109/TIT.2009.2021376.

[6]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory, 57 (2011), 1165-1173.  doi: 10.1109/TIT.2010.2095232.

[7]

É. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredachi Inf., 21 (1985), 3-16. 

[8]

É. M. Gabidulin and M. Bossert, Codes for network coding, Problemy Peredachi Informatsii, 45 (2009), 54-68; translation in Probl. Inf. Transm., 45 (2009), 343-356 doi: 10.1134/S003294600904005X.

[9]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.

[10]

E. Gorla and A. Ravagnani, Subspace codes from Ferrers diagrams, J. Algebra. Appl, 16 (2017), 1750131, 23 pp. doi: 10.1142/S0219498817501316.

[11]

X. HeY. ChenZ. Zhang and K. Zhou, New construction for constant dimension subspace codes via a composite structure, IEEE Communications Letters, 25 (2021), 1422-1426.  doi: 10.1109/LCOMM.2021.3052734.

[12]

D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inf. Theory, 67 (2021), 705-715.  doi: 10.1109/TIT.2020.3038272.

[13]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, arXiv preprint, arXiv: 1601.02864.

[14]

D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, Coding theory and applications, Lecture Notes in Comput. Sci., Springer, Cham, 10495 (2017), 163-191. doi: 10.1007/978-3-319-66278-7_15.

[15]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.

[16]

S. Kurz, Lifted codes and the multilevel construction for constant dimension codes, arXiv preprint, arXiv: 2004.14241.

[17]

S. Kurz, The interplay of different metrics for the construction of constant dimension codes, arXiv preprint, arXiv: 2109.07128. doi: 10.3934/amc.2021069.

[18]

H. LaoH. Chen and X. Tan, New constant dimension subspace codes from block inserting constructions, Cryptography and Communications, 14 (2022), 87-99.  doi: 10.1007/s12095-021-00524-9.

[19]

F. Li, Construction of constant dimension subspace codes by modifying linkage construction, IEEE Trans. Inf. Theory, 66 (2019), 2760-2764.  doi: 10.1109/TIT.2019.2960343.

[20]

S. LiuY. Chang and T. Feng, Parallel multilevel constructions for constant dimension codes, IEEE Trans. Inf. Theory, 66 (2020), 6884-6897.  doi: 10.1109/TIT.2020.3004315.

[21]

Y. NiuQ. Yue and D. Huang, New constant dimension subspace codes from parallel linkage construction and multilevel construction, Cryptogr. Commun., 14 (2021), 201-214.  doi: 10.1007/s12095-021-00504-z.

[22]

Y. NiuQ. Yue and D. Huang, New constant dimension subspace codes from generalized inserting construction, IEEE Communications Letters, 25 (2020), 1066-1069.  doi: 10.1109/LCOMM.2020.3046042.

[23]

Y. Niu, Q. Yue and D. Huang, Construction of constant dimension codes via improved inserting construction, Applicable Algebra in Engineering, Communication and Computing, 1–18. doi: 10.1007/s00200-021-00537-0.

[24]

N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, ferrers diagrams, and pending blocks, IEEE Trans. Inf. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.

[25]

D. SilvaF. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inf. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.

[26]

A. TrautmannJ. Rosenthal and A. Edelmayer, New improvements on the echelon-ferrers construction, Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, 19 (2010), 405-408. 

[27]

L. Xu and H. Chen, New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319.  doi: 10.1109/TIT.2018.2839596.

Table 1.  New lower bounds of CDCs and comparisons
$ \mathcal{{A}}_q(n,d,k) $ New Old
$ \mathcal{{A}}_2(10,4,5) $ 1182 792 1179 355 [17]
$ \mathcal{{A}}_2(14,4,7) $ 4980 3800 62528 4980 1091 73760 [18]
$ \mathcal{{A}}_2(18,6,9) $ 9271 5451 7995 6957 680 9271 5451 5683 7632 063 [21]
$ \mathcal{{A}}_2(19,4,9) $ 1368 9705 2676 1712 8565 96480 1348 0021 4626 1417 4474 06857 [16]
$ \mathcal{{A}}_2(19,6,9) $ 1185 6367 8468 1892 4462 08 1183 0716 7986 1259 2217 64 [13]
$ \mathcal{{A}}_q(n,d,k) $ New Old
$ \mathcal{{A}}_2(10,4,5) $ 1182 792 1179 355 [17]
$ \mathcal{{A}}_2(14,4,7) $ 4980 3800 62528 4980 1091 73760 [18]
$ \mathcal{{A}}_2(18,6,9) $ 9271 5451 7995 6957 680 9271 5451 5683 7632 063 [21]
$ \mathcal{{A}}_2(19,4,9) $ 1368 9705 2676 1712 8565 96480 1348 0021 4626 1417 4474 06857 [16]
$ \mathcal{{A}}_2(19,6,9) $ 1185 6367 8468 1892 4462 08 1183 0716 7986 1259 2217 64 [13]
Table 2.  New lower bounds of CDCs with $ q = 3,4,5,7,8,9 $ and comparisons
$ \mathcal{A}_q(n,d,k) $ New Old
$\mathcal{A}_3(10,4,5)$ 3554987040 3554758008 [2]
$\mathcal{A}_4(10,4,5)$ 1105477214240 1105472134720 [2]
$\mathcal{A}_5(10,4,5)$ 95563890394800 95563833229500 [2]
$\mathcal{A}_7(10,4,5)$ 79831697457099392 79831695230704816 [2]
$\mathcal{A}_8(10,4,5)$ 1153247498650857600 1153247489101767168 [2]
$\mathcal{A}_9(10,4,5)$ 12159772626275657280 12159772591745929140 [2]
$\mathcal{A}_3(14,4,7)$ 111580711538886863118 111580699486197627621 [18]
$\mathcal{A}_4(14,4,7)$ 19448126131859716486238208 19448126106855252931117056 [18]
$\mathcal{A}_5(14,4,7)$ 227842765470062186386570093750 227842765460471817497177734375 [18]
$\mathcal{A}_7(14,4,7)$ 312127712852708781630023191664496798 312127712852630817229550525046012105 [18]
$\mathcal{A}_8(14,4,7)$ 85094651546423058721849360550392496128 85094651546420258527817077896654094336 [18]
$\mathcal{A}_9(14,4,7)$ 11974590568455135712446431471768531529318 11974590568455069588288269079025246897611 [18]
$\mathcal{A}_3(18,6,9)$ 1144661280188122797462569962566 1144661280188113244283258523275 [21]
$\mathcal{A}_4(18,6,9)$ 85071058146182803382564529885253322496 85071058146182803276539661530357567488 [21]
$\mathcal{A}_5(18,6,9)$ 108420289965710977906843346838623736530883750 1084202899657109779066908466204627308207046882 [21]
$\mathcal{A}_7(18,6,9)$ 174251503388975551318884931829216767244210120069652870 174251503388975551318884922599501083082039017444351219 [21]
$\mathcal{A}_8(18,6,9)$ 784637723721919791138381635372340213678508318750302990336 784637723721919791138381634635240574256149776594272256000 [21]
$\mathcal{A}_9(18,6,9)$ 1310020512493866339206870302364431654473106442742625891884486 1310020512493866339206870302329188829727612166983542933136923 [21]
$\mathcal{A}_3(19,4,9)$ 150729716220941500233452603175442841600 150024595081884393007012600338236544880 [16]
$\mathcal{A}_4(19,4,9)$ 1469459223991290829562363939740408362925117407232 1467749485360498225634923971135876693953433631137 [16]
$\mathcal{A}_5(19,4,9)$ 82888717036604453802134197600449240321151282440598192128 82857418316609426935359109104035114463034 035070083878426 [16]
$\mathcal{A}_7(19,4,9)$ 4055625553923365061772296535671638571873481904180965005209
0342408192
4055349977189813448098217800392083438240002715009770943419
9076454452 [16]
$\mathcal{A}_8(19,4,9)$ 17673467678334608788939021059103044220727521426926854164420
72047684681728
17672860099364199843916651708091969078559219854089318997680
42741180322945 [16]
$\mathcal{A}_9(19,4,9)$ 21851237218522468409236985988907225047508253380547793759917
936643985041784832
21850825132503488679494454968297144272741734751304289430201
060161991357689262 [16]
$\mathcal{A}_3(19,6,9)$ 2503331918360606794969179010105344 2503282684929477428119980988170240 [13]
$\mathcal{A}_4(19,6,9)$ 1393802887884996972927002540806573603684352 1393801891840429495613125677692710966788096 [13]
$\mathcal{A}_5(19,6,9)$ 8470334286280497166922325808103328806034003197952 8470333809353914153414266564962173680273118461952 [13]
$\mathcal{A}_7(19,6,9)$ 143503605347450349088548148205477322768334792113472690716672 1435036051660215437893875356369795515110086873
11855223308288 [13]
$\mathcal{A}_8(19,6,9)$ 1645504570046382782137926420946189395893170277923947298487795712 1645504569581170369082080049424126722185209334
583492840817426432 [13]
$\mathcal{A}_9(19,6,9)$ 6265787498825269087366283145948755008928094349595662198570215
276544
6265787498351064233039139483141141083293979708221
793185986684911616 [13]
$ \mathcal{A}_q(n,d,k) $ New Old
$\mathcal{A}_3(10,4,5)$ 3554987040 3554758008 [2]
$\mathcal{A}_4(10,4,5)$ 1105477214240 1105472134720 [2]
$\mathcal{A}_5(10,4,5)$ 95563890394800 95563833229500 [2]
$\mathcal{A}_7(10,4,5)$ 79831697457099392 79831695230704816 [2]
$\mathcal{A}_8(10,4,5)$ 1153247498650857600 1153247489101767168 [2]
$\mathcal{A}_9(10,4,5)$ 12159772626275657280 12159772591745929140 [2]
$\mathcal{A}_3(14,4,7)$ 111580711538886863118 111580699486197627621 [18]
$\mathcal{A}_4(14,4,7)$ 19448126131859716486238208 19448126106855252931117056 [18]
$\mathcal{A}_5(14,4,7)$ 227842765470062186386570093750 227842765460471817497177734375 [18]
$\mathcal{A}_7(14,4,7)$ 312127712852708781630023191664496798 312127712852630817229550525046012105 [18]
$\mathcal{A}_8(14,4,7)$ 85094651546423058721849360550392496128 85094651546420258527817077896654094336 [18]
$\mathcal{A}_9(14,4,7)$ 11974590568455135712446431471768531529318 11974590568455069588288269079025246897611 [18]
$\mathcal{A}_3(18,6,9)$ 1144661280188122797462569962566 1144661280188113244283258523275 [21]
$\mathcal{A}_4(18,6,9)$ 85071058146182803382564529885253322496 85071058146182803276539661530357567488 [21]
$\mathcal{A}_5(18,6,9)$ 108420289965710977906843346838623736530883750 1084202899657109779066908466204627308207046882 [21]
$\mathcal{A}_7(18,6,9)$ 174251503388975551318884931829216767244210120069652870 174251503388975551318884922599501083082039017444351219 [21]
$\mathcal{A}_8(18,6,9)$ 784637723721919791138381635372340213678508318750302990336 784637723721919791138381634635240574256149776594272256000 [21]
$\mathcal{A}_9(18,6,9)$ 1310020512493866339206870302364431654473106442742625891884486 1310020512493866339206870302329188829727612166983542933136923 [21]
$\mathcal{A}_3(19,4,9)$ 150729716220941500233452603175442841600 150024595081884393007012600338236544880 [16]
$\mathcal{A}_4(19,4,9)$ 1469459223991290829562363939740408362925117407232 1467749485360498225634923971135876693953433631137 [16]
$\mathcal{A}_5(19,4,9)$ 82888717036604453802134197600449240321151282440598192128 82857418316609426935359109104035114463034 035070083878426 [16]
$\mathcal{A}_7(19,4,9)$ 4055625553923365061772296535671638571873481904180965005209
0342408192
4055349977189813448098217800392083438240002715009770943419
9076454452 [16]
$\mathcal{A}_8(19,4,9)$ 17673467678334608788939021059103044220727521426926854164420
72047684681728
17672860099364199843916651708091969078559219854089318997680
42741180322945 [16]
$\mathcal{A}_9(19,4,9)$ 21851237218522468409236985988907225047508253380547793759917
936643985041784832
21850825132503488679494454968297144272741734751304289430201
060161991357689262 [16]
$\mathcal{A}_3(19,6,9)$ 2503331918360606794969179010105344 2503282684929477428119980988170240 [13]
$\mathcal{A}_4(19,6,9)$ 1393802887884996972927002540806573603684352 1393801891840429495613125677692710966788096 [13]
$\mathcal{A}_5(19,6,9)$ 8470334286280497166922325808103328806034003197952 8470333809353914153414266564962173680273118461952 [13]
$\mathcal{A}_7(19,6,9)$ 143503605347450349088548148205477322768334792113472690716672 1435036051660215437893875356369795515110086873
11855223308288 [13]
$\mathcal{A}_8(19,6,9)$ 1645504570046382782137926420946189395893170277923947298487795712 1645504569581170369082080049424126722185209334
583492840817426432 [13]
$\mathcal{A}_9(19,6,9)$ 6265787498825269087366283145948755008928094349595662198570215
276544
6265787498351064233039139483141141083293979708221
793185986684911616 [13]
[1]

Shuhui Yu, Lijun Ji, Shuangqing Liu. Bilateral multilevel construction of constant dimension codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022056

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