# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021069
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## 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.  Google Scholar [2] M. Braun, P. 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.  Google Scholar [3] H. Chen, X. He, J. 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.  Google Scholar [4] A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, Advances in Mathematics of Communications, (to appear), 15 pp. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] T. Feng, S. Kurz and S. Liu, Bounds for the multilevel construction, arXiv preprint, arXiv: 2011.06937, (2020), 95 pp. Google Scholar [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.  Google Scholar [11] H. Gluesing-Luerssen, K. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] X. He, Y. 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.  Google Scholar [16] X. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] T. Honold, M. 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.  Google Scholar [24] T. Honold, M. 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.  Google Scholar [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.  Google Scholar [26] S. Kurz, A note on the linkage construction for constant dimension codes, arXiv preprint, arXiv: 1906.09780, (2019), 13 pp. Google Scholar [27] S. Kurz, Generalized LMRD code bounds for constant dimension codes, IEEE Communications Letters, 24 (2020), 2100-2103.  doi: 10.1109/LCOMM.2020.3003132.  Google Scholar [28] S. Kurz, Lifted codes and the multilevel construction for constant dimension codes, arXiv preprint, arXiv: 2004.14241, (2020), 40 pp. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [32] S. Liu, Y. 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.  Google Scholar [33] S. Liu, Y. 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.  Google Scholar [34] Y. Niu, Q. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] M. Braun, P. 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.  Google Scholar [3] H. Chen, X. He, J. 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.  Google Scholar [4] A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, Advances in Mathematics of Communications, (to appear), 15 pp. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] T. Feng, S. Kurz and S. Liu, Bounds for the multilevel construction, arXiv preprint, arXiv: 2011.06937, (2020), 95 pp. Google Scholar [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.  Google Scholar [11] H. Gluesing-Luerssen, K. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] X. He, Y. 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.  Google Scholar [16] X. He, Y. Chen, Z. 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.  Google Scholar [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.  Google Scholar [18] D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, arXiv preprint, arXiv: 1601.02864, (2016), 44 pp. Google Scholar [19] D. Heinlein, M. Kiermaier, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] T. Honold, M. 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.  Google Scholar [24] T. Honold, M. 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.  Google Scholar [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.  Google Scholar [26] S. Kurz, A note on the linkage construction for constant dimension codes, arXiv preprint, arXiv: 1906.09780, (2019), 13 pp. Google Scholar [27] S. Kurz, Generalized LMRD code bounds for constant dimension codes, IEEE Communications Letters, 24 (2020), 2100-2103.  doi: 10.1109/LCOMM.2020.3003132.  Google Scholar [28] S. Kurz, Lifted codes and the multilevel construction for constant dimension codes, arXiv preprint, arXiv: 2004.14241, (2020), 40 pp. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [32] S. Liu, Y. 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.  Google Scholar [33] S. Liu, Y. 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.  Google Scholar [34] Y. Niu, Q. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 Google Scholar [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.  Google Scholar [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.  Google Scholar
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 $i$
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 $m(q, \mathcal{F},1)/m(q, \mathcal{F},2)$ $11000$ $q^3$ $q^3$ $10100$ $q^2$ $q^3$ $10010$ $q$ $q^3$ $10001$ $1$ $q^3$ $01100$ $q^2$ $q^2$ $01010$ $q$ $q^2$ $01001$ $1$ $q^2$ $00110$ $1$ $q^2$ $00101$ $1$ $q$ $00011$ $1$ $1$
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 $\{11000,00110\}$ $q^3+1$ $q^2$ $\{11000,00101\}$ $q^3+1$ $q$ $\{11000,00011\}$ $q^3+1$ $1$ $\{11000\}$ $q^3$ $q^3-q^2-q-1$ $\{10100,01010\}$ $q^2+q$ $q^2$ $\{10100,01001\}$ $q^2+1$ $q^2$ $\{10100\}$ $q^2$ $q^3-2q^2$ $\{01100,10010\}$ $q^2+q$ $q^2$ $\{10010\}$ $q$ $q^3-q^2$ $\{10001\}$ $1$ $q^3$
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