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Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $
Orbit codes from forms on vector spaces over a finite field
1. | Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, I-70126, Italy |
2. | Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Potenza, I-85100, Italy |
3. | Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, I-80138, Italy |
In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an $ n $-dimensional vector space over the finite field $ {\mathbb F_{q}} $. All these codes admit the general linear group $ {{{{\rm{GL}}}}}(n,q) $ as a transitive automorphism group.
References:
[1] |
R. Ahlswede, N. Cai, S.-Y. Li and R. W. Yeung,
Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.
doi: 10.1109/18.850663. |
[2] |
M. Aschbacher, Finite Group Theory, Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 1986.
![]() |
[3] |
E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv,
Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165.
doi: 10.1109/TIT.2016.2520479. |
[4] |
O. Bottema, On the Betti-Mathieu group, Nieuw Arch. Wisk., 16 (1930), 46-50. Google Scholar |
[5] |
M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of $q$–analogs of Steiner systems, Forum Math. Pi, 4 (2016), e7, 14 pp.
doi: 10.1017/fmp.2016.5. |
[6] |
L. Carlitz, A Note on the Betti-Mathieu group, Portugaliae mathematica, 22 (1963), 121-125. Google Scholar |
[7] |
B. Chen and H. Liu,
Constructions of cyclic constant dimension codes, Des. Codes Cryptogr., 86 (2018), 1267-1279.
doi: 10.1007/s10623-017-0394-9. |
[8] |
J.-J. Climent, V. Requena and X. Soler-Escrivà,
A construction of Abelian non-cyclic orbit codes, Cryptography and Communication, 11 (2019), 839-852.
doi: 10.1007/s12095-018-0306-5. |
[9] |
B. N. Cooperstein,
External flats to varieties in ${{{\rm PG}}}(M_{n, n}({{{\rm GF}}}(q)))$, Linear Algebra Appl., 267 (1997), 175-186.
|
[10] |
A. Cossidente and F. Pavese,
On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.
doi: 10.1007/s10623-014-0018-6. |
[11] |
A. Cossidente and F. Pavese,
Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.
doi: 10.1007/s10623-015-0166-3. |
[12] |
A. Cossidente and F. Pavese,
Subspace codes in ${{{\rm PG}}}(2n-1, q)$, Combinatorica, 37 (2017), 1073-1095.
doi: 10.1007/s00493-016-3354-5. |
[13] |
A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, 107–129, Signals Commun. Technol., Springer, Cham, 2018. |
[14] |
A. Cossidente, F. Pavese and L. Storme,
Optimal subspace codes in ${{{\rm PG}}}(4, q)$, Adv. Math. Commun., 13 (2019), 393-404.
doi: 10.3934/amc.2019025. |
[15] |
A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, preprint, arXiv: 1911.03387. Google Scholar |
[16] |
B. Csajbók and A. Siciliano,
Puncturing maximum rank distance codes, J. Algebraic Combin., 49 (2019), 507-534.
doi: 10.1007/s10801-018-0833-3. |
[17] |
P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-New York, 1968. |
[18] |
N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Combin., 24 (2017), 18 pp. |
[19] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank- metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[20] |
T. Etzion and N. Silberstein,
Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.
doi: 10.1109/TIT.2012.2220119. |
[21] |
T. Etzion and A. Vardy,
Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[22] |
G. Faina, G. Kiss, S. Marcugini and F. Pambianco,
The cyclic model for ${{{\rm PG}}}(n-1, q)$ and a construction of arcs, European J. Combin., 23 (2002), 31-35.
doi: 10.1006/eujc.2001.0525. |
[23] |
H. Gluesing-Luerssen, K. Morrison and C. Troha,
Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197.
doi: 10.3934/amc.2015.9.177. |
[24] |
H. Gluesing-Luerssen and C. Troha,
Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.
doi: 10.3934/amc.2016023. |
[25] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864, 2016. Google Scholar |
[26] |
J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd ed, Clarendon Press, Oxford, 1998.
![]() |
[27] |
T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi and B. Leong,
A random linear network coding approach to multicast, IEEE Trans. Inform. Theory, 52 (2006), 4413-4430.
doi: 10.1109/TIT.2006.881746. |
[28] |
T. Ho, R. Koetter, M. Médard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in Proceedings of the 2003 IEEE international symposium on information theory (ISIT 2003), Yokohama, Japan. IEEE, (2003), p442.
doi: 10.1109/ISIT.2003.1228459. |
[29] |
T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, 131–170, Signals Commun. Technol., Springer, Cham, 2018. |
[30] |
A.-L. Horlemann-Trautmann,
Message encoding and retrieval for spread and cyclic orbit codes, Des. Codes Cryptogr., 86 (2018), 365-386.
doi: 10.1007/s10623-017-0377-x. |
[31] |
A. L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, 25–42, Signals Commun. Technol., Springer, Cham, 2018. |
[32] |
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-New York, 1967. |
[33] |
W. M. Kantor,
Linear groups containing a Singer cycle, J. Algebra, 62 (1980), 232-234.
doi: 10.1016/0021-8693(80)90214-8. |
[34] |
B. C. Kestenband,
Finite projective geometries that are incidence structures of caps, Linear Algebra Appl., 48 (1982), 303-313.
doi: 10.1016/0024-3795(82)90116-1. |
[35] |
A. Kohnert and S. Kurz,
Construction of large constant-dimension codes with a prescribed minimum distance, Lecture Notes in Computer Science, 5393 (2008), 31-42.
doi: 10.1007/978-3-540-89994-5_4. |
[36] |
R. Kötter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[37] |
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
![]() |
[38] |
K. Otal and F. Özbudak,
Cyclic subspace codes via subspace polynomials, Des. Codes Cryptogr., 85 (2017), 191-204.
doi: 10.1007/s10623-016-0297-1. |
[39] |
K. Otal and F. Özbudak, Constructions of cyclic subspace codes and maximum rank distance codes, in Network Coding and Subspace Designs, 43–66, Signals Commun. Technol., Springer, Cham, 2018. |
[40] |
M. H. Poroch and A. A. Talebi, Product of symplectic groups and its cyclic orbit code, Discrete Math. Algorithms Appl., 11 (2019), 1950061, 25 pp.
doi: 10.1142/s1793830919500617. |
[41] |
N. Silberstein and A.-L. Trautmann,
Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.
doi: 10.1109/TIT.2015.2435743. |
[42] |
D. Silva, F. R. Kschischang and R. Koetter,
A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[43] |
J. Singer,
A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.
doi: 10.1090/S0002-9947-1938-1501951-4. |
[44] |
A.-L. Trautmann,
Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.
doi: 10.3934/amc.2013.7.147. |
[45] |
A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal,
Cyclic orbit codes, IEEE Trans. Inf. Theory, 59 (2013), 7386-7404.
doi: 10.1109/TIT.2013.2274266. |
[46] |
A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in Proc. IEEE Inf. Theory Workshop, Dublin, Ireland, 2010, 1–4.
doi: 10.1109/CIG.2010.5592788. |
[47] |
D. E. Taylor, The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9. Heldermann Verlag, Berlin, 1992. |
[48] |
Z.-X. Wan, Geometry of matrices, World Scientific Publishing Co. NJ, 1996.
doi: 10.1142/9789812830234. |
[49] |
B. Wu and Z. Liu,
Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.
doi: 10.1016/j.ffa.2013.03.003. |
[50] |
S.-T. Xia and F.-W. Fu,
Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.
doi: 10.1007/s10623-008-9221-7. |
show all references
References:
[1] |
R. Ahlswede, N. Cai, S.-Y. Li and R. W. Yeung,
Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.
doi: 10.1109/18.850663. |
[2] |
M. Aschbacher, Finite Group Theory, Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 1986.
![]() |
[3] |
E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv,
Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165.
doi: 10.1109/TIT.2016.2520479. |
[4] |
O. Bottema, On the Betti-Mathieu group, Nieuw Arch. Wisk., 16 (1930), 46-50. Google Scholar |
[5] |
M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of $q$–analogs of Steiner systems, Forum Math. Pi, 4 (2016), e7, 14 pp.
doi: 10.1017/fmp.2016.5. |
[6] |
L. Carlitz, A Note on the Betti-Mathieu group, Portugaliae mathematica, 22 (1963), 121-125. Google Scholar |
[7] |
B. Chen and H. Liu,
Constructions of cyclic constant dimension codes, Des. Codes Cryptogr., 86 (2018), 1267-1279.
doi: 10.1007/s10623-017-0394-9. |
[8] |
J.-J. Climent, V. Requena and X. Soler-Escrivà,
A construction of Abelian non-cyclic orbit codes, Cryptography and Communication, 11 (2019), 839-852.
doi: 10.1007/s12095-018-0306-5. |
[9] |
B. N. Cooperstein,
External flats to varieties in ${{{\rm PG}}}(M_{n, n}({{{\rm GF}}}(q)))$, Linear Algebra Appl., 267 (1997), 175-186.
|
[10] |
A. Cossidente and F. Pavese,
On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.
doi: 10.1007/s10623-014-0018-6. |
[11] |
A. Cossidente and F. Pavese,
Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.
doi: 10.1007/s10623-015-0166-3. |
[12] |
A. Cossidente and F. Pavese,
Subspace codes in ${{{\rm PG}}}(2n-1, q)$, Combinatorica, 37 (2017), 1073-1095.
doi: 10.1007/s00493-016-3354-5. |
[13] |
A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, 107–129, Signals Commun. Technol., Springer, Cham, 2018. |
[14] |
A. Cossidente, F. Pavese and L. Storme,
Optimal subspace codes in ${{{\rm PG}}}(4, q)$, Adv. Math. Commun., 13 (2019), 393-404.
doi: 10.3934/amc.2019025. |
[15] |
A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, preprint, arXiv: 1911.03387. Google Scholar |
[16] |
B. Csajbók and A. Siciliano,
Puncturing maximum rank distance codes, J. Algebraic Combin., 49 (2019), 507-534.
doi: 10.1007/s10801-018-0833-3. |
[17] |
P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-New York, 1968. |
[18] |
N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Combin., 24 (2017), 18 pp. |
[19] |
T. Etzion and N. Silberstein,
Error-correcting codes in projective spaces via rank- metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.
doi: 10.1109/TIT.2009.2021376. |
[20] |
T. Etzion and N. Silberstein,
Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.
doi: 10.1109/TIT.2012.2220119. |
[21] |
T. Etzion and A. Vardy,
Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.
doi: 10.1109/TIT.2010.2095232. |
[22] |
G. Faina, G. Kiss, S. Marcugini and F. Pambianco,
The cyclic model for ${{{\rm PG}}}(n-1, q)$ and a construction of arcs, European J. Combin., 23 (2002), 31-35.
doi: 10.1006/eujc.2001.0525. |
[23] |
H. Gluesing-Luerssen, K. Morrison and C. Troha,
Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197.
doi: 10.3934/amc.2015.9.177. |
[24] |
H. Gluesing-Luerssen and C. Troha,
Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.
doi: 10.3934/amc.2016023. |
[25] |
D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864, 2016. Google Scholar |
[26] |
J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd ed, Clarendon Press, Oxford, 1998.
![]() |
[27] |
T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi and B. Leong,
A random linear network coding approach to multicast, IEEE Trans. Inform. Theory, 52 (2006), 4413-4430.
doi: 10.1109/TIT.2006.881746. |
[28] |
T. Ho, R. Koetter, M. Médard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in Proceedings of the 2003 IEEE international symposium on information theory (ISIT 2003), Yokohama, Japan. IEEE, (2003), p442.
doi: 10.1109/ISIT.2003.1228459. |
[29] |
T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, 131–170, Signals Commun. Technol., Springer, Cham, 2018. |
[30] |
A.-L. Horlemann-Trautmann,
Message encoding and retrieval for spread and cyclic orbit codes, Des. Codes Cryptogr., 86 (2018), 365-386.
doi: 10.1007/s10623-017-0377-x. |
[31] |
A. L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, 25–42, Signals Commun. Technol., Springer, Cham, 2018. |
[32] |
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-New York, 1967. |
[33] |
W. M. Kantor,
Linear groups containing a Singer cycle, J. Algebra, 62 (1980), 232-234.
doi: 10.1016/0021-8693(80)90214-8. |
[34] |
B. C. Kestenband,
Finite projective geometries that are incidence structures of caps, Linear Algebra Appl., 48 (1982), 303-313.
doi: 10.1016/0024-3795(82)90116-1. |
[35] |
A. Kohnert and S. Kurz,
Construction of large constant-dimension codes with a prescribed minimum distance, Lecture Notes in Computer Science, 5393 (2008), 31-42.
doi: 10.1007/978-3-540-89994-5_4. |
[36] |
R. Kötter and F. R. Kschischang,
Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[37] |
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
![]() |
[38] |
K. Otal and F. Özbudak,
Cyclic subspace codes via subspace polynomials, Des. Codes Cryptogr., 85 (2017), 191-204.
doi: 10.1007/s10623-016-0297-1. |
[39] |
K. Otal and F. Özbudak, Constructions of cyclic subspace codes and maximum rank distance codes, in Network Coding and Subspace Designs, 43–66, Signals Commun. Technol., Springer, Cham, 2018. |
[40] |
M. H. Poroch and A. A. Talebi, Product of symplectic groups and its cyclic orbit code, Discrete Math. Algorithms Appl., 11 (2019), 1950061, 25 pp.
doi: 10.1142/s1793830919500617. |
[41] |
N. Silberstein and A.-L. Trautmann,
Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.
doi: 10.1109/TIT.2015.2435743. |
[42] |
D. Silva, F. R. Kschischang and R. Koetter,
A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[43] |
J. Singer,
A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.
doi: 10.1090/S0002-9947-1938-1501951-4. |
[44] |
A.-L. Trautmann,
Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.
doi: 10.3934/amc.2013.7.147. |
[45] |
A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal,
Cyclic orbit codes, IEEE Trans. Inf. Theory, 59 (2013), 7386-7404.
doi: 10.1109/TIT.2013.2274266. |
[46] |
A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in Proc. IEEE Inf. Theory Workshop, Dublin, Ireland, 2010, 1–4.
doi: 10.1109/CIG.2010.5592788. |
[47] |
D. E. Taylor, The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9. Heldermann Verlag, Berlin, 1992. |
[48] |
Z.-X. Wan, Geometry of matrices, World Scientific Publishing Co. NJ, 1996.
doi: 10.1142/9789812830234. |
[49] |
B. Wu and Z. Liu,
Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.
doi: 10.1016/j.ffa.2013.03.003. |
[50] |
S.-T. Xia and F.-W. Fu,
Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.
doi: 10.1007/s10623-008-9221-7. |
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