November  2020, 14(4): 631-650. doi: 10.3934/amc.2020035

Abelian non-cyclic orbit codes and multishot subspace codes

1. 

Department of Mathematics and Statistics, Federal University of São João del-Rei, Praça Frei Orlando, 170, Centro, São João del-Rei - MG, 36307-352, Brazil

2. 

Department of Communications, FEEC, State University of Campinas, Cidade Universitária Zeferino Vaz - Barão Geraldo, Campinas - SP, 13083-852, Brazil

3. 

Department of Mathematics, Federal University of Viçosa, Avenida Peter Henry Rolfs, s/n, Viçosa - MG, 36570-900, Brazil

Received  November 2018 Revised  June 2019 Published  November 2020 Early access  November 2019

Fund Project: The first author was supported by CAPES and CNPq PhD scholarships

In this paper we characterize the orbit codes as geometrically uniform codes. This characterization is based on the description of all isometries over a projective geometry. In addition, Abelian orbit codes are defined and a construction of Abelian non-cyclic orbit codes is presented. In order to analyze their structures, the concept of geometrically uniform partitions have to be reinterpreted. As a consequence, a substantial reduction in the number of computations needed to obtain the minimum subspace distance of these codes is achieved and established.

An application of orbit codes to multishot subspace codes obtained according to a multi-level construction is provided.

Citation: 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
References:
[1]

R. AhlswedeN. CaiR. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.  doi: 10.1109/18.850663.

[2]

R. Baer, Linear Algebra and Projective Geometry, Academic Press Inc., New York, 1952.

[3]

F. Bardestani and A. Iranmanesh, Cyclic orbit codes with the normalizer of a singer subgroup, J. Sci. Islam. Repub. Iran, 26 (2015), 49-55. 

[4]

E. Biglieri and M. Elia, Multidimensional modulation and coding for band-limited digital channels, IEEE Trans. Inform. Theory, 34 (1988), 803-809.  doi: 10.1109/18.9777.

[5]

A. R. Calderbank, Multilevel codes and multistage decoding, IEEE Trans. Comm., 37 (1989), 222-229.  doi: 10.1109/26.20095.

[6]

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.

[7]

J.-J. ClimentV. Requena and X. S.-Escrivà, A construction of Abelian non-cyclic orbit codes, Cryptogr. Commun., 11 (2019), 839-852.  doi: 10.1007/s12095-018-0306-5.

[8]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Fundamental Principles of Mathematical Sciences, 290, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7.

[9]

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

[10]

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.

[11]

G. Forney Jr., Geometrically uniform codes, IEEE Trans. Inform. Theory, 37 (1991), 1241-1260.  doi: 10.1109/18.133243.

[12]

È. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16. 

[13]

J. T. Goozeff, Abelian p-subgroups of the general linear group, J. Austral. Math. Soc., 11 (1970), 257-259.  doi: 10.1017/S1446788700006613.

[14]

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

[15]

H. Imai and S. Hirakawa, A new multilevel coding method using error-correcting codes, IEEE Trans. Inform. Theory, 23 (1977), 371-377.  doi: 10.1109/TIT.1977.1055718.

[16]

I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/092.

[17]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in Cryptography and Coding, Lecture Notes in Comput. Sci., 5921, Springer, Berlin, 2009, 1-21. doi: 10.1007/978-3-642-10868-6_1.

[18]

R. Köetter 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.

[19]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, Lecture Notes in Comput. Sci., 5393, Springer, Berlin, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4.

[20]

H. A. Loeliger, Signal sets matched to groups, IEEE Trans. Inform. Theory, 37 (1991), 1675-1682.  doi: 10.1109/18.104333.

[21]

H. G-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.

[22]

F. ManganielloE. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding, IEEE International Symposium on Information Theory, (2008), 881-885.  doi: 10.1109/ISIT.2008.4595113.

[23]

R. W. Nobrega and B. F. Uchoa-Filho, Multishot codes for network coding: Bounds and a multilevel construction, IEEE International Symposium on Information Theory, (2009), 428-432.  doi: 10.1109/ISIT.2009.5205750.

[24]

J. J. Rotman, An Introduction to the Theory of Groups, Graduate Texts in Mathematics, 148, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.

[25]

D. Slepian, Group codes for the Gaussian channel, Bell System Tech. J., 47 (1968), 575-602.  doi: 10.1002/j.1538-7305.1968.tb02486.x.

[26]

D. A. Suprunenko, Matrix Groups, Translations of Mathematical Monographs, 45, American Mathematical Society, Providence, RI, 1976.

[27]

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

[28]

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.

[29]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - A new concept in the area of network coding, IEEE Information Theory Workshop, Dublin, Ireland, 2010, 1-4. doi: 10.1109/CIG.2010.5592788.

[30]

Z. Wan, On geometrically uniform signal sets and signal sets matched to groups, IEEE International Symposium on Information Theory, San Antonio, 1993,179pp. doi: 10.1109/ISIT.1993.748493.

show all references

References:
[1]

R. AhlswedeN. CaiR. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.  doi: 10.1109/18.850663.

[2]

R. Baer, Linear Algebra and Projective Geometry, Academic Press Inc., New York, 1952.

[3]

F. Bardestani and A. Iranmanesh, Cyclic orbit codes with the normalizer of a singer subgroup, J. Sci. Islam. Repub. Iran, 26 (2015), 49-55. 

[4]

E. Biglieri and M. Elia, Multidimensional modulation and coding for band-limited digital channels, IEEE Trans. Inform. Theory, 34 (1988), 803-809.  doi: 10.1109/18.9777.

[5]

A. R. Calderbank, Multilevel codes and multistage decoding, IEEE Trans. Comm., 37 (1989), 222-229.  doi: 10.1109/26.20095.

[6]

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.

[7]

J.-J. ClimentV. Requena and X. S.-Escrivà, A construction of Abelian non-cyclic orbit codes, Cryptogr. Commun., 11 (2019), 839-852.  doi: 10.1007/s12095-018-0306-5.

[8]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Fundamental Principles of Mathematical Sciences, 290, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7.

[9]

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

[10]

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.

[11]

G. Forney Jr., Geometrically uniform codes, IEEE Trans. Inform. Theory, 37 (1991), 1241-1260.  doi: 10.1109/18.133243.

[12]

È. M. Gabidulin, Theory of codes with maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16. 

[13]

J. T. Goozeff, Abelian p-subgroups of the general linear group, J. Austral. Math. Soc., 11 (1970), 257-259.  doi: 10.1017/S1446788700006613.

[14]

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

[15]

H. Imai and S. Hirakawa, A new multilevel coding method using error-correcting codes, IEEE Trans. Inform. Theory, 23 (1977), 371-377.  doi: 10.1109/TIT.1977.1055718.

[16]

I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/092.

[17]

A. Khaleghi, D. Silva and F. R. Kschischang, Subspace codes, in Cryptography and Coding, Lecture Notes in Comput. Sci., 5921, Springer, Berlin, 2009, 1-21. doi: 10.1007/978-3-642-10868-6_1.

[18]

R. Köetter 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.

[19]

A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical Methods in Computer Science, Lecture Notes in Comput. Sci., 5393, Springer, Berlin, 2008, 31-42. doi: 10.1007/978-3-540-89994-5_4.

[20]

H. A. Loeliger, Signal sets matched to groups, IEEE Trans. Inform. Theory, 37 (1991), 1675-1682.  doi: 10.1109/18.104333.

[21]

H. G-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.

[22]

F. ManganielloE. Gorla and J. Rosenthal, Spread codes and spread decoding in network coding, IEEE International Symposium on Information Theory, (2008), 881-885.  doi: 10.1109/ISIT.2008.4595113.

[23]

R. W. Nobrega and B. F. Uchoa-Filho, Multishot codes for network coding: Bounds and a multilevel construction, IEEE International Symposium on Information Theory, (2009), 428-432.  doi: 10.1109/ISIT.2009.5205750.

[24]

J. J. Rotman, An Introduction to the Theory of Groups, Graduate Texts in Mathematics, 148, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.

[25]

D. Slepian, Group codes for the Gaussian channel, Bell System Tech. J., 47 (1968), 575-602.  doi: 10.1002/j.1538-7305.1968.tb02486.x.

[26]

D. A. Suprunenko, Matrix Groups, Translations of Mathematical Monographs, 45, American Mathematical Society, Providence, RI, 1976.

[27]

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

[28]

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.

[29]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - A new concept in the area of network coding, IEEE Information Theory Workshop, Dublin, Ireland, 2010, 1-4. doi: 10.1109/CIG.2010.5592788.

[30]

Z. Wan, On geometrically uniform signal sets and signal sets matched to groups, IEEE International Symposium on Information Theory, San Antonio, 1993,179pp. doi: 10.1109/ISIT.1993.748493.

Table 1.  Interdistance sets $ D\left( {\left\{ V \right\},{C_H}\left( {{\alpha ^i}V} \right)} \right)$, for $ 1 \leq i \leq 4 $
$ d_S (.,.) $ $ \alpha^1 $ $ \alpha^{10} $ $ \alpha^{19} $ $ \alpha^{28} $ $ \alpha^{37} $ $ \alpha^{46} $ $ \alpha^{55} $
$ \alpha^0 $ 4 4 6 6 6 4 6
$ d_S (.,.) $ $ \alpha^2 $ $ \alpha^{11} $ $ \alpha^{20} $ $ \alpha^{29} $ $ \alpha^{38} $ $ \alpha^{47} $ $ \alpha^{56} $
$ \alpha^0 $ 4 6 4 4 6 4 6
$ d_S (.,.) $ $ \alpha^3 $ $ \alpha^{12} $ $ \alpha^{21} $ $ \alpha^{30} $ $ \alpha^{39} $ $ \alpha^{48} $ $ \alpha^{57} $
$ \alpha^0 $ 4 4 6 4 4 4 4
$ d_S (.,.) $ $ \alpha^4 $ $ \alpha^{13} $ $ \alpha^{22} $ $ \alpha^{31} $ $ \alpha^{40} $ $ \alpha^{49} $ $ \alpha^{58} $
$ \alpha^0 $ 4 6 6 4 4 6 4
$ d_S (.,.) $ $ \alpha^1 $ $ \alpha^{10} $ $ \alpha^{19} $ $ \alpha^{28} $ $ \alpha^{37} $ $ \alpha^{46} $ $ \alpha^{55} $
$ \alpha^0 $ 4 4 6 6 6 4 6
$ d_S (.,.) $ $ \alpha^2 $ $ \alpha^{11} $ $ \alpha^{20} $ $ \alpha^{29} $ $ \alpha^{38} $ $ \alpha^{47} $ $ \alpha^{56} $
$ \alpha^0 $ 4 6 4 4 6 4 6
$ d_S (.,.) $ $ \alpha^3 $ $ \alpha^{12} $ $ \alpha^{21} $ $ \alpha^{30} $ $ \alpha^{39} $ $ \alpha^{48} $ $ \alpha^{57} $
$ \alpha^0 $ 4 4 6 4 4 4 4
$ d_S (.,.) $ $ \alpha^4 $ $ \alpha^{13} $ $ \alpha^{22} $ $ \alpha^{31} $ $ \alpha^{40} $ $ \alpha^{49} $ $ \alpha^{58} $
$ \alpha^0 $ 4 6 6 4 4 6 4
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