doi: 10.3934/amc.2021069
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

The interplay of different metrics for the construction of constant dimension codes

Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

Received  August 2021 Revised  November 2021 Early access January 2022

A basic problem for constant dimension codes is to determine the maximum possible size $ A_q(n,d;k) $ of a set of $ k $-dimensional subspaces in $ \mathbb{F}_q^n $, called codewords, such that the subspace distance satisfies $ d_S(U,W): = 2k-2\dim(U\cap W)\ge d $ for all pairs of different codewords $ U $, $ W $. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for $ A_q(n,d;k) $ are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show the potential for further improvements. As examples we give improved constructions for the cases $ A_q(10,4;5) $, $ A_q(11,4;4) $, $ A_q(12,6;6) $, and $ A_q(15,4;4) $. We also derive general upper bounds for subcodes arising in those constructions.

Citation: Sascha Kurz. The interplay of different metrics for the construction of constant dimension codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021069
References:
[1]

J. Antrobus and H. Gluesing-Luerssen, Maximal Ferrers diagram codes: Constructions and genericity considerations, IEEE Transactions on Information Theory, 65 (2019), 6204-6223.  doi: 10.1109/TIT.2019.2926256.

[2]

M. BraunP. R. J. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Experimental Mathematics, 27 (2018), 179-183.  doi: 10.1080/10586458.2016.1239145.

[3]

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

[4]

A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, Advances in Mathematics of Communications, (to appear), 15 pp.

[5]

A. Cossidente and F. Pavese, Subspace codes in PG(2n-1, q), Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.

[6]

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

[7]

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

[8]

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

[9]

T. Feng, S. Kurz and S. Liu, Bounds for the multilevel construction, arXiv preprint, arXiv: 2011.06937, (2020), 95 pp.

[10]

M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Transactions on Information Theory, 56 (2010), 3207-3216.  doi: 10.1109/TIT.2010.2048447.

[11]

H. Gluesing-LuerssenK. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Advances in Mathematics of Communications, 9 (2015), 177-197.  doi: 10.3934/amc.2015.9.177.

[12]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Advances in Mathematics of Communications, 10 (2016), 525-540.  doi: 10.3934/amc.2016023.

[13]

M. Greferath, M. O. Pavčevi$\acute{\rm c} $, N. Silberstein and M. Á. Vázquez-Castro, Network Coding and Subspace Designs, Springer, 2018. doi: 10.1007/978-3-319-70293-3.

[14]

X. He, Construction of constant dimension codes from two parallel versions of linkage construction, IEEE Communications Letters, 24 (2020), 2392-2395.  doi: 10.1109/LCOMM.2020.3012488.

[15]

X. HeY. Chen and Z. Zhang, Improving the linkage construction with Echelon-Ferrers for constant-dimension codes, IEEE Communications Letters, 24 (2020), 1875-1879.  doi: 10.1109/LCOMM.2020.2997928.

[16]

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.

[17]

D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Transactions on Information Theory, 67 (2020), 705-715.  doi: 10.1109/TIT.2020.3038272.

[18]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, arXiv preprint, arXiv: 1601.02864, (2016), 44 pp.

[19]

D. HeinleinM. KiermaierS. Kurz and A. Wassermann, A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Advances in Mathematics of Communications, 13 (2019), 457-475.  doi: 10.3934/amc.2019029.

[20]

D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in International Castle Meeting on Coding Theory and Applications, Springer, 2017,163–191. doi: 10.1007/978-3-319-66278-7_15.

[21]

D. Heinlein and S. Kurz, Coset construction for subspace codes, IEEE Transactions on Information Theory, 63 (2017), 7651-7660.  doi: 10.1109/TIT.2017.2753822.

[22]

T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday, World Scientific, 2016,141–175.

[23]

T. HonoldM. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, Contemp. Math., 632 (2015), 157-176.  doi: 10.1090/conm/632/12627.

[24]

T. HonoldM. Kiermaier and S. Kurz, Classification of large partial plane spreads in PG(6, 2) and related combinatorial objects, Journal of Geometry, 110 (2019), 1-31.  doi: 10.1007/s00022-018-0459-6.

[25]

M. Kiermaier and S. Kurz, On the lengths of divisible codes, IEEE Transactions on Information Theory, 66 (2020), 4051-4060.  doi: 10.1109/TIT.2020.2968832.

[26]

S. Kurz, A note on the linkage construction for constant dimension codes, arXiv preprint, arXiv: 1906.09780, (2019), 13 pp.

[27]

S. Kurz, Generalized LMRD code bounds for constant dimension codes, IEEE Communications Letters, 24 (2020), 2100-2103.  doi: 10.1109/LCOMM.2020.3003132.

[28]

S. Kurz, Lifted codes and the multilevel construction for constant dimension codes, arXiv preprint, arXiv: 2004.14241, (2020), 40 pp.

[29]

S. Kurz, Subspaces intersecting in at most a point, Designs, Codes and Cryptography, 88 (2020), 595-599.  doi: 10.1007/s10623-019-00699-6.

[30]

H. Lao, H. Chen, J. Weng and X. Tan, Parameter-controlled inserting constructions of constant dimension subspace codes, arXiv preprint, arXiv: 2008.09944, (2020), 48 pp.

[31]

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

[32]

S. LiuY. Chang and T. Feng, Constructions for optimal Ferrers diagram rank-metric codes, IEEE Transactions on Information Theory, 65 (2019), 4115-4130.  doi: 10.1109/TIT.2019.2894401.

[33]

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

[34]

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.

[35]

J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics doi: 10.1515/9783110642094-013.

[36]

N. Silberstein and T. Etzion, Large constant dimension codes and lexicodes, Advances in Mathematics of Communications, 5 (2011), 177-189.  doi: 10.3934/amc.2011.5.177.

[37]

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

[38]

A.-L. Trautmann and J. Rosenthal, New improvements on the Echelon-Ferrers construction, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems–MTNS, vol. 5.9, 2010, 405-408

[39]

S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Designs, Codes and Cryptography, 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.

[40]

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

show all references

References:
[1]

J. Antrobus and H. Gluesing-Luerssen, Maximal Ferrers diagram codes: Constructions and genericity considerations, IEEE Transactions on Information Theory, 65 (2019), 6204-6223.  doi: 10.1109/TIT.2019.2926256.

[2]

M. BraunP. R. J. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Experimental Mathematics, 27 (2018), 179-183.  doi: 10.1080/10586458.2016.1239145.

[3]

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

[4]

A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, Advances in Mathematics of Communications, (to appear), 15 pp.

[5]

A. Cossidente and F. Pavese, Subspace codes in PG(2n-1, q), Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.

[6]

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

[7]

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

[8]

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

[9]

T. Feng, S. Kurz and S. Liu, Bounds for the multilevel construction, arXiv preprint, arXiv: 2011.06937, (2020), 95 pp.

[10]

M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Transactions on Information Theory, 56 (2010), 3207-3216.  doi: 10.1109/TIT.2010.2048447.

[11]

H. Gluesing-LuerssenK. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Advances in Mathematics of Communications, 9 (2015), 177-197.  doi: 10.3934/amc.2015.9.177.

[12]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Advances in Mathematics of Communications, 10 (2016), 525-540.  doi: 10.3934/amc.2016023.

[13]

M. Greferath, M. O. Pavčevi$\acute{\rm c} $, N. Silberstein and M. Á. Vázquez-Castro, Network Coding and Subspace Designs, Springer, 2018. doi: 10.1007/978-3-319-70293-3.

[14]

X. He, Construction of constant dimension codes from two parallel versions of linkage construction, IEEE Communications Letters, 24 (2020), 2392-2395.  doi: 10.1109/LCOMM.2020.3012488.

[15]

X. HeY. Chen and Z. Zhang, Improving the linkage construction with Echelon-Ferrers for constant-dimension codes, IEEE Communications Letters, 24 (2020), 1875-1879.  doi: 10.1109/LCOMM.2020.2997928.

[16]

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.

[17]

D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Transactions on Information Theory, 67 (2020), 705-715.  doi: 10.1109/TIT.2020.3038272.

[18]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, arXiv preprint, arXiv: 1601.02864, (2016), 44 pp.

[19]

D. HeinleinM. KiermaierS. Kurz and A. Wassermann, A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Advances in Mathematics of Communications, 13 (2019), 457-475.  doi: 10.3934/amc.2019029.

[20]

D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in International Castle Meeting on Coding Theory and Applications, Springer, 2017,163–191. doi: 10.1007/978-3-319-66278-7_15.

[21]

D. Heinlein and S. Kurz, Coset construction for subspace codes, IEEE Transactions on Information Theory, 63 (2017), 7651-7660.  doi: 10.1109/TIT.2017.2753822.

[22]

T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday, World Scientific, 2016,141–175.

[23]

T. HonoldM. Kiermaier and S. Kurz, Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, Contemp. Math., 632 (2015), 157-176.  doi: 10.1090/conm/632/12627.

[24]

T. HonoldM. Kiermaier and S. Kurz, Classification of large partial plane spreads in PG(6, 2) and related combinatorial objects, Journal of Geometry, 110 (2019), 1-31.  doi: 10.1007/s00022-018-0459-6.

[25]

M. Kiermaier and S. Kurz, On the lengths of divisible codes, IEEE Transactions on Information Theory, 66 (2020), 4051-4060.  doi: 10.1109/TIT.2020.2968832.

[26]

S. Kurz, A note on the linkage construction for constant dimension codes, arXiv preprint, arXiv: 1906.09780, (2019), 13 pp.

[27]

S. Kurz, Generalized LMRD code bounds for constant dimension codes, IEEE Communications Letters, 24 (2020), 2100-2103.  doi: 10.1109/LCOMM.2020.3003132.

[28]

S. Kurz, Lifted codes and the multilevel construction for constant dimension codes, arXiv preprint, arXiv: 2004.14241, (2020), 40 pp.

[29]

S. Kurz, Subspaces intersecting in at most a point, Designs, Codes and Cryptography, 88 (2020), 595-599.  doi: 10.1007/s10623-019-00699-6.

[30]

H. Lao, H. Chen, J. Weng and X. Tan, Parameter-controlled inserting constructions of constant dimension subspace codes, arXiv preprint, arXiv: 2008.09944, (2020), 48 pp.

[31]

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

[32]

S. LiuY. Chang and T. Feng, Constructions for optimal Ferrers diagram rank-metric codes, IEEE Transactions on Information Theory, 65 (2019), 4115-4130.  doi: 10.1109/TIT.2019.2894401.

[33]

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

[34]

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.

[35]

J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics doi: 10.1515/9783110642094-013.

[36]

N. Silberstein and T. Etzion, Large constant dimension codes and lexicodes, Advances in Mathematics of Communications, 5 (2011), 177-189.  doi: 10.3934/amc.2011.5.177.

[37]

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

[38]

A.-L. Trautmann and J. Rosenthal, New improvements on the Echelon-Ferrers construction, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems–MTNS, vol. 5.9, 2010, 405-408

[39]

S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Designs, Codes and Cryptography, 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.

[40]

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

Table 1.  Notation and abbreviations
CDC constant dimension code
MRD maximum rank distance
FDRM Ferrers diagram rank metric
$ d_S(U,W) $ subspace distance between codewords $ U $ and $ W $
$ d_S( \mathcal{C}) $ minimum subspace distance of a CDC $ \mathcal{C} $
$ d_H(u,w) $ Hamming distance between codewords $ u $ and $ w $
$ d_H( \mathcal{S}) $ minimum Hamming distance of $ \mathcal{S} $
$ d_H( \mathcal{V}, \mathcal{V}') $ minimum Hamming distance between a codeword in $ \mathcal{V} $ and
a codeword in $ \mathcal{V}' $
$ d_R(A,B) $ rank distance between two matrices $ A $ and $ B $
$ \operatorname{rk}(A) $ rank of a matrix $ A $
$ \mathcal{G}_1(n,k) $ set of binary vectors of length $ n $ and Hamming weight $ k $
$ \mathcal{G}_q(n,k) $ set of $ k $-dimensional subspaces in $ \mathbb{F}_q^n $
$\left[\begin{array}{l} n \\ k \end{array}\right]_{q} $ Gaussian binomial coefficient; $ \# \mathcal{G}_q(n,k) $
$ A_q(n,d;k) $ maximum possible cardinality of a CDC $ \mathcal{C}\subseteq \mathcal{G}_{q}(n,k) $
with minimum subspace distance at least $ d $
$ m(q,m,n, d_R) $ number of codewords of an $ (m\times n, d_R)_q $-MRD code
$ a(q,m,n, d_R,r) $ number of codewords of rank $ r $ in an additive $ (m\times n, d_R)_q $-MRD code
$ E(U) $, $ E(M) $ matrix $ M $ or generator matrix of $ U $ in reduced row echelon form
$ v(U) $, $ v(M) $ pivot vector
$ {n_1 \choose k_1},\dots, {n_l \choose k_l} $ set of binary vectors
$ T(U) $ Ferrers tableaux
$ \mathcal{F}(U) $, $ \mathcal{F}(v) $ Ferrers diagram
$ A_q(n,d;k; \mathcal{V}) $ max. possible cardinality of a CDC $ \mathcal{C}\subseteq \mathcal{G}_{q}(n,k) $ with min.
subspace distance at least $ d $ whose codewords have pivot vectors in $ \mathcal{V} $
$ I_k $ $ k\times k $ unit matrix
$ e_i $ unit vector with a one at position $ i $
CDC constant dimension code
MRD maximum rank distance
FDRM Ferrers diagram rank metric
$ d_S(U,W) $ subspace distance between codewords $ U $ and $ W $
$ d_S( \mathcal{C}) $ minimum subspace distance of a CDC $ \mathcal{C} $
$ d_H(u,w) $ Hamming distance between codewords $ u $ and $ w $
$ d_H( \mathcal{S}) $ minimum Hamming distance of $ \mathcal{S} $
$ d_H( \mathcal{V}, \mathcal{V}') $ minimum Hamming distance between a codeword in $ \mathcal{V} $ and
a codeword in $ \mathcal{V}' $
$ d_R(A,B) $ rank distance between two matrices $ A $ and $ B $
$ \operatorname{rk}(A) $ rank of a matrix $ A $
$ \mathcal{G}_1(n,k) $ set of binary vectors of length $ n $ and Hamming weight $ k $
$ \mathcal{G}_q(n,k) $ set of $ k $-dimensional subspaces in $ \mathbb{F}_q^n $
$\left[\begin{array}{l} n \\ k \end{array}\right]_{q} $ Gaussian binomial coefficient; $ \# \mathcal{G}_q(n,k) $
$ A_q(n,d;k) $ maximum possible cardinality of a CDC $ \mathcal{C}\subseteq \mathcal{G}_{q}(n,k) $
with minimum subspace distance at least $ d $
$ m(q,m,n, d_R) $ number of codewords of an $ (m\times n, d_R)_q $-MRD code
$ a(q,m,n, d_R,r) $ number of codewords of rank $ r $ in an additive $ (m\times n, d_R)_q $-MRD code
$ E(U) $, $ E(M) $ matrix $ M $ or generator matrix of $ U $ in reduced row echelon form
$ v(U) $, $ v(M) $ pivot vector
$ {n_1 \choose k_1},\dots, {n_l \choose k_l} $ set of binary vectors
$ T(U) $ Ferrers tableaux
$ \mathcal{F}(U) $, $ \mathcal{F}(v) $ Ferrers diagram
$ A_q(n,d;k; \mathcal{V}) $ max. possible cardinality of a CDC $ \mathcal{C}\subseteq \mathcal{G}_{q}(n,k) $ with min.
subspace distance at least $ d $ whose codewords have pivot vectors in $ \mathcal{V} $
$ I_k $ $ k\times k $ unit matrix
$ e_i $ unit vector with a one at position $ i $
Table 2.  Data for Lemma 3.4 with $ \mathcal{F}\in \mathcal{G}_1(5,2) $
pivot vector size $ m(q, \mathcal{F},2) $ $ \# $ of cosets $ m(q, \mathcal{F},1)/m(q, \mathcal{F},2) $
$ 11000 $ $ q^3 $ $ q^3 $
$ 10100 $ $ q^2 $ $ q^3 $
$ 10010 $ $ q $ $ q^3 $
$ 10001 $ $ 1 $ $ q^3 $
$ 01100 $ $ q^2 $ $ q^2 $
$ 01010 $ $ q $ $ q^2 $
$ 01001 $ $ 1 $ $ q^2 $
$ 00110 $ $ 1 $ $ q^2 $
$ 00101 $ $ 1 $ $ q $
$ 00011 $ $ 1 $ $ 1 $
pivot vector size $ m(q, \mathcal{F},2) $ $ \# $ of cosets $ m(q, \mathcal{F},1)/m(q, \mathcal{F},2) $
$ 11000 $ $ q^3 $ $ q^3 $
$ 10100 $ $ q^2 $ $ q^3 $
$ 10010 $ $ q $ $ q^3 $
$ 10001 $ $ 1 $ $ q^3 $
$ 01100 $ $ q^2 $ $ q^2 $
$ 01010 $ $ q $ $ q^2 $
$ 01001 $ $ 1 $ $ q^2 $
$ 00110 $ $ 1 $ $ q^2 $
$ 00101 $ $ 1 $ $ q $
$ 00011 $ $ 1 $ $ 1 $
Table 3.  Packing scheme for Proposition 3.7
skeleton code size $ \# $ of used cosets
$ \{11000,00110\} $ $ q^3+1 $ $ q^2 $
$ \{11000,00101\} $ $ q^3+1 $ $ q $
$ \{11000,00011\} $ $ q^3+1 $ $ 1 $
$ \{11000\} $ $ q^3 $ $ q^3-q^2-q-1 $
$ \{10100,01010\} $ $ q^2+q $ $ q^2 $
$ \{10100,01001\} $ $ q^2+1 $ $ q^2 $
$ \{10100\} $ $ q^2 $ $ q^3-2q^2 $
$ \{01100,10010\} $ $ q^2+q $ $ q^2 $
$ \{10010\} $ $ q $ $ q^3-q^2 $
$ \{10001\} $ $ 1 $ $ q^3 $
skeleton code size $ \# $ of used cosets
$ \{11000,00110\} $ $ q^3+1 $ $ q^2 $
$ \{11000,00101\} $ $ q^3+1 $ $ q $
$ \{11000,00011\} $ $ q^3+1 $ $ 1 $
$ \{11000\} $ $ q^3 $ $ q^3-q^2-q-1 $
$ \{10100,01010\} $ $ q^2+q $ $ q^2 $
$ \{10100,01001\} $ $ q^2+1 $ $ q^2 $
$ \{10100\} $ $ q^2 $ $ q^3-2q^2 $
$ \{01100,10010\} $ $ q^2+q $ $ q^2 $
$ \{10010\} $ $ q $ $ q^3-q^2 $
$ \{10001\} $ $ 1 $ $ q^3 $
[1]

Huimin Lao, Hao Chen. New constant dimension subspace codes from multilevel linkage construction. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022039

[2]

Thomas Honold, Michael Kiermaier, Sascha Kurz. Constructions and bounds for mixed-dimension subspace codes. Advances in Mathematics of Communications, 2016, 10 (3) : 649-682. doi: 10.3934/amc.2016033

[3]

Antonio Cossidente, Sascha Kurz, Giuseppe Marino, Francesco Pavese. Combining subspace codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021007

[4]

Heide Gluesing-Luerssen, Carolyn Troha. Construction of subspace codes through linkage. Advances in Mathematics of Communications, 2016, 10 (3) : 525-540. doi: 10.3934/amc.2016023

[5]

Ernst M. Gabidulin, Pierre Loidreau. Properties of subspace subcodes of Gabidulin codes. Advances in Mathematics of Communications, 2008, 2 (2) : 147-157. doi: 10.3934/amc.2008.2.147

[6]

Daniel Heinlein, Ferdinand Ihringer. New and updated semidefinite programming bounds for subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 613-630. doi: 10.3934/amc.2020034

[7]

Daniel Heinlein, Sascha Kurz. Binary subspace codes in small ambient spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 817-839. doi: 10.3934/amc.2018048

[8]

Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035

[9]

Anna-Lena Trautmann. Isometry and automorphisms of constant dimension codes. Advances in Mathematics of Communications, 2013, 7 (2) : 147-160. doi: 10.3934/amc.2013.7.147

[10]

Natalia Silberstein, Tuvi Etzion. Large constant dimension codes and lexicodes. Advances in Mathematics of Communications, 2011, 5 (2) : 177-189. doi: 10.3934/amc.2011.5.177

[11]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055

[12]

John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019

[13]

Andries E. Brouwer, Tuvi Etzion. Some new distance-4 constant weight codes. Advances in Mathematics of Communications, 2011, 5 (3) : 417-424. doi: 10.3934/amc.2011.5.417

[14]

Antonio Cossidente, Francesco Pavese, Leo Storme. Optimal subspace codes in $ {{\rm{PG}}}(4,q) $. Advances in Mathematics of Communications, 2019, 13 (3) : 393-404. doi: 10.3934/amc.2019025

[15]

Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046

[16]

Zihui Liu. Galois LCD codes over rings. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022002

[17]

Daniele Bartoli, Matteo Bonini, Massimo Giulietti. Constant dimension codes from Riemann-Roch spaces. Advances in Mathematics of Communications, 2017, 11 (4) : 705-713. doi: 10.3934/amc.2017051

[18]

Tatsuya Maruta, Yusuke Oya. On optimal ternary linear codes of dimension 6. Advances in Mathematics of Communications, 2011, 5 (3) : 505-520. doi: 10.3934/amc.2011.5.505

[19]

Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249

[20]

Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273

2021 Impact Factor: 1.015

Article outline

Figures and Tables

[Back to Top]