doi: 10.3934/amc.2021040
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Automorphism groups and isometries for cyclic orbit codes

Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA

* Corresponding author: Heide-Gluesing Luerssen

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: The first author was partially supported by the grant #422479 from the Simons Foundation

We study orbit codes in the field extension $ \mathbb{F}_{q^n} $. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of $ \mathbb{F}_{q^n} $. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

Citation: Heide Gluesing-Luerssen, Hunter Lehmann. Automorphism groups and isometries for cyclic orbit codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021040
References:
[1]

E. Ben-SassonT. EtzionA. Gabizon and N. Raviv, Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165.  doi: 10.1109/TIT.2016.2520479.  Google Scholar

[2]

M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wasserman, Existence of $q$-analogs of Steiner systems, Forum of Mathematics, Pi, 4 (2016). doi: 10.1017/fmp.2016.5.  Google Scholar

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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.  Google Scholar

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A. Cossidente and M. J. de Resmini, Remarks on Singer cylic groups and their normalizers, Des. Codes Cryptogr., 32 (2004), 97-102.  doi: 10.1023/B:DESI.0000029214.50635.17.  Google Scholar

[5]

K. Drudge, On the orbits of {S}inger groups and their subgroups, Electron. J. Combin., 9 (2002), Research Paper 15, 10 pp. doi: 10.37236/1632.  Google Scholar

[6]

A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding, In Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, Hungary, 2010, 1811–1814. Google Scholar

[7]

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.  Google Scholar

[8]

N. Gill, On a conjecture of Degos, Cah. Topol. Géom. Différ. Catég., 57 (2016), 229-237.   Google Scholar

[9]

H. Gluesing-Luerssen and H. Lehmann, Distance distributions of cyclic orbit codes, Des. Codes Cryptogr., 89 (2021), 447-470.  doi: 10.1007/s10623-020-00823-x.  Google Scholar

[10]

H. Gluesing-LuerssenK. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197.  doi: 10.3934/amc.2015.9.177.  Google Scholar

[11]

J. Gomez-Calderon, On the stabilizer of companion matrices, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 140-143.   Google Scholar

[12]

M. D. Hestenes, Singer groups, Canadian Journal of Mathematics, 22 (1970), 492-513.  doi: 10.4153/CJM-1970-057-2.  Google Scholar

[13]

B. Huppert, Endliche Gruppen, Springer, Berlin, Heidelberg, New York, 1967.  Google Scholar

[14]

W. M. Kantor, Linear groups containing a Singer cycle, Journal of Algebra, 62 (1980), 232-234.  doi: 10.1016/0021-8693(80)90214-8.  Google Scholar

[15]

R. Koetter 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.  Google Scholar

[16]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, In J. Calmet, W. Geiselmann, and J. M{ü}ller-Quade, editors, Mathematical Methods in Computer Science, volume 5393. Lecture Notes in Computer Science; Springer, Berlin, 2008, 31–42. doi: 10.1007/978-3-540-89994-5_4.  Google Scholar

[17]

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.  Google Scholar

[18]

R. M. RothN. Raviv and I. Tamo, Construction of Sidon spaces with applications to coding, IEEE Trans. Inform. Theory, 64 (2018), 4412-4422.  doi: 10.1109/TIT.2017.2766178.  Google Scholar

[19]

A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.  doi: 10.3934/amc.2013.7.147.  Google Scholar

[20]

A.-L. TrautmannF. ManganielloM. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inform. Theory, 59 (2013), 7386-7404.  doi: 10.1109/TIT.2013.2274266.  Google Scholar

[21]

W. Zhao and X. Tang, A characterization of cyclic subspace codes via subspace polynomials, Finite Fields Appl., 57 (2019), 1-12.  doi: 10.1016/j.ffa.2019.01.002.  Google Scholar

show all references

References:
[1]

E. Ben-SassonT. EtzionA. Gabizon and N. Raviv, Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165.  doi: 10.1109/TIT.2016.2520479.  Google Scholar

[2]

M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wasserman, Existence of $q$-analogs of Steiner systems, Forum of Mathematics, Pi, 4 (2016). doi: 10.1017/fmp.2016.5.  Google Scholar

[3]

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.  Google Scholar

[4]

A. Cossidente and M. J. de Resmini, Remarks on Singer cylic groups and their normalizers, Des. Codes Cryptogr., 32 (2004), 97-102.  doi: 10.1023/B:DESI.0000029214.50635.17.  Google Scholar

[5]

K. Drudge, On the orbits of {S}inger groups and their subgroups, Electron. J. Combin., 9 (2002), Research Paper 15, 10 pp. doi: 10.37236/1632.  Google Scholar

[6]

A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding, In Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, Hungary, 2010, 1811–1814. Google Scholar

[7]

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.  Google Scholar

[8]

N. Gill, On a conjecture of Degos, Cah. Topol. Géom. Différ. Catég., 57 (2016), 229-237.   Google Scholar

[9]

H. Gluesing-Luerssen and H. Lehmann, Distance distributions of cyclic orbit codes, Des. Codes Cryptogr., 89 (2021), 447-470.  doi: 10.1007/s10623-020-00823-x.  Google Scholar

[10]

H. Gluesing-LuerssenK. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197.  doi: 10.3934/amc.2015.9.177.  Google Scholar

[11]

J. Gomez-Calderon, On the stabilizer of companion matrices, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 140-143.   Google Scholar

[12]

M. D. Hestenes, Singer groups, Canadian Journal of Mathematics, 22 (1970), 492-513.  doi: 10.4153/CJM-1970-057-2.  Google Scholar

[13]

B. Huppert, Endliche Gruppen, Springer, Berlin, Heidelberg, New York, 1967.  Google Scholar

[14]

W. M. Kantor, Linear groups containing a Singer cycle, Journal of Algebra, 62 (1980), 232-234.  doi: 10.1016/0021-8693(80)90214-8.  Google Scholar

[15]

R. Koetter 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.  Google Scholar

[16]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, In J. Calmet, W. Geiselmann, and J. M{ü}ller-Quade, editors, Mathematical Methods in Computer Science, volume 5393. Lecture Notes in Computer Science; Springer, Berlin, 2008, 31–42. doi: 10.1007/978-3-540-89994-5_4.  Google Scholar

[17]

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.  Google Scholar

[18]

R. M. RothN. Raviv and I. Tamo, Construction of Sidon spaces with applications to coding, IEEE Trans. Inform. Theory, 64 (2018), 4412-4422.  doi: 10.1109/TIT.2017.2766178.  Google Scholar

[19]

A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.  doi: 10.3934/amc.2013.7.147.  Google Scholar

[20]

A.-L. TrautmannF. ManganielloM. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inform. Theory, 59 (2013), 7386-7404.  doi: 10.1109/TIT.2013.2274266.  Google Scholar

[21]

W. Zhao and X. Tang, A characterization of cyclic subspace codes via subspace polynomials, Finite Fields Appl., 57 (2019), 1-12.  doi: 10.1016/j.ffa.2019.01.002.  Google Scholar

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