American Institute of Mathematical Sciences

Advanced
August  2021, 15(3): 423-440. doi: 10.3934/amc.2020075

$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $

Dean Crnković 1, , Nina Mostarac 1, , Bernardo G. Rodrigues 2, and  Leo Storme 3,

1. 

Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia
2. 

Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0002, Pretoria, South Africa
3. 

Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Ghent, Belgium

Received  September 2019 Published  August 2021 Early access  April 2020

In this paper we construct $ 2 $-PD-sets of $ 16 $ elements for codes from the Desarguesian projective planes $ \mathrm{PG}(2,q) $, where $ q = 2^h $ and $ 5\leq h \leq 9 $. We also construct $ 3 $-PD-sets of $ 75 $ elements for the code from the Desarguesian projective plane $ \mathrm{PG}(2,q) $, where $ q = 2^9 $. These $ 2 $-PD-sets and $ 3 $-PD-sets can be used for partial permutation decoding of codes obtained from the Desarguesian projective planes.

Keywords: Desarguesian projective plane, code, PD-set, s-PD-set, permutation decoding.
Mathematics Subject Classification: Primary: 51E20, 94B05.
Citation: Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075
References:
[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar
[2]

D. Crnković and N. Mostarac, PD-sets for codes related to flag-transitive symmetric designs, Trans. Comb., 7 (2018), 37-50.  doi: 10.22108/toc.2017.21615.  Google Scholar

[3]

D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543.  doi: 10.1109/TIT.1982.1056504.  Google Scholar

[4]

J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[5]

W. C. Huffman, Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2 (1998), 1345-1440.   Google Scholar

[6]

J. D. Key, Permutation decoding for codes from designs, finite geometries and graphs, Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS, Amsterdam, 29 (2011), 172-201.   Google Scholar

[7]

J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682.  doi: 10.1016/j.ejc.2004.04.007.  Google Scholar

[8]

J. MacWilliams, Permutation decoding of systematic codes, Bell Syst. Tech. J., 43 (1964), 485-505.  doi: 10.1002/j.1538-7305.1964.tb04075.x.  Google Scholar

[9]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[10]

G. E. Moorhouse, Bruck nets, codes, and characters of loops, Des. Codes Cryptogr., 1 (1991), 7-29.  doi: 10.1007/BF00123956.  Google Scholar

[11]

N. Pace and A. Sonnino, On linear codes admitting large automorphism groups, Des. Codes Cryptogr., 83 (2017), 115-143.  doi: 10.1007/s10623-016-0207-6.  Google Scholar

[12]

K. J. C. Smith, On the $p$-rank of the incidence matrix of points in hyperplanes in a finite projective geometry, J. Combin. Theory, 7 (1969), 122-129.  doi: 10.1016/S0021-9800(69)80046-3.  Google Scholar

[13]

P. Vandendriessche, Codes of Desarguesian projective planes of even order, projective triads and $(q + t, t)$-arcs of type $(0, 2, t)$, Finite Fields Appl., 17 (2011), 521-531.  doi: 10.1016/j.ffa.2011.03.003.  Google Scholar

[14]

P. Vandendriessche, Intertwined Results on Linear Codes and Galois Geometries, Ph.D thesis, Ghent University, Faculty of Sciences, Ghent, Belgium, 2014. https://cage.ugent.be/geometry/theses.php. Google Scholar

show all references

References:
[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar
[2]

D. Crnković and N. Mostarac, PD-sets for codes related to flag-transitive symmetric designs, Trans. Comb., 7 (2018), 37-50.  doi: 10.22108/toc.2017.21615.  Google Scholar

[3]

D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543.  doi: 10.1109/TIT.1982.1056504.  Google Scholar

[4]

J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[5]

W. C. Huffman, Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2 (1998), 1345-1440.   Google Scholar

[6]

J. D. Key, Permutation decoding for codes from designs, finite geometries and graphs, Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS, Amsterdam, 29 (2011), 172-201.   Google Scholar

[7]

J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682.  doi: 10.1016/j.ejc.2004.04.007.  Google Scholar

[8]

J. MacWilliams, Permutation decoding of systematic codes, Bell Syst. Tech. J., 43 (1964), 485-505.  doi: 10.1002/j.1538-7305.1964.tb04075.x.  Google Scholar

[9]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[10]

G. E. Moorhouse, Bruck nets, codes, and characters of loops, Des. Codes Cryptogr., 1 (1991), 7-29.  doi: 10.1007/BF00123956.  Google Scholar

[11]

N. Pace and A. Sonnino, On linear codes admitting large automorphism groups, Des. Codes Cryptogr., 83 (2017), 115-143.  doi: 10.1007/s10623-016-0207-6.  Google Scholar

[12]

K. J. C. Smith, On the $p$-rank of the incidence matrix of points in hyperplanes in a finite projective geometry, J. Combin. Theory, 7 (1969), 122-129.  doi: 10.1016/S0021-9800(69)80046-3.  Google Scholar

[13]

P. Vandendriessche, Codes of Desarguesian projective planes of even order, projective triads and $(q + t, t)$-arcs of type $(0, 2, t)$, Finite Fields Appl., 17 (2011), 521-531.  doi: 10.1016/j.ffa.2011.03.003.  Google Scholar

[14]

P. Vandendriessche, Intertwined Results on Linear Codes and Galois Geometries, Ph.D thesis, Ghent University, Faculty of Sciences, Ghent, Belgium, 2014. https://cage.ugent.be/geometry/theses.php. Google Scholar

Table 1.  Codes of $ \mathrm{PG}(2,q) $: lower bounds on sizes of PD-sets and $ 2 $-PD-sets
$ q $ Code $ t $ $ r $ $ b $ $ b_2 $
$ 32 $ [1057,244, 33] 16 813 180 3
$ 64 $ [4161,730, 65] 32 3431 1623 3
$ 128 $ [16513, 2188,129] 64 14325 40696 3
$ 256 $ [65793, 6562,257] 128 59231 3965945 3
$ 512 $ [262657, 19684,513] 256 242973 3625171287 3
Table Options

Download as excel

$ q $ Code $ t $ $ r $ $ b $ $ b_2 $
$ 32 $ [1057,244, 33] 16 813 180 3
$ 64 $ [4161,730, 65] 32 3431 1623 3
$ 128 $ [16513, 2188,129] 64 14325 40696 3
$ 256 $ [65793, 6562,257] 128 59231 3965945 3
$ 512 $ [262657, 19684,513] 256 242973 3625171287 3
[1]

Anna-Lena Horlemann-Trautmann, Violetta Weger. Information set decoding in the Lee metric with applications to cryptography. Advances in Mathematics of Communications, 2021, 15 (4) : 677-699. doi: 10.3934/amc.2020089
[2]

Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
[3]

Urszula Ledzewicz, Heinz Schättler. The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 561-578. doi: 10.3934/mbe.2005.2.561
[4]

Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065
[5]

Valeria Artale, Cristina L. R. Milazzo, Calogero Orlando, Angela Ricciardello. Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2067-2091. doi: 10.3934/jimo.2017032
[6]

Jonas Eriksson. A weight-based characterization of the set of correctable error patterns under list-of-2 decoding. Advances in Mathematics of Communications, 2007, 1 (3) : 331-356. doi: 10.3934/amc.2007.1.331
[7]

Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[8]

François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[9]

Luis A. Fernández, Cecilia Pola. Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1563-1588. doi: 10.3934/dcdsb.2014.19.1563
[10]

Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008
[11]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[12]

Tiago Carvalho, Luiz Fernando Gonçalves. A flow on $ S^2 $ presenting the ball as its minimal set. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4263-4280. doi: 10.3934/dcdsb.2020287
[13]

Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[14]

Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[15]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[16]

Sanjit Chatterjee, Chethan Kamath, Vikas Kumar. Private set-intersection with common set-up. Advances in Mathematics of Communications, 2018, 12 (1) : 17-47. doi: 10.3934/amc.2018002
[17]

Víctor Jiménez López, Gabriel Soler López. A topological characterization of ω-limit sets for continuous flows on the projective plane. Conference Publications, 2001, 2001 (Special) : 254-258. doi: 10.3934/proc.2001.2001.254
[18]

Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165
[19]

Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2291-2303. doi: 10.3934/cpaa.2012.11.2291
[20]

Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, Katsutoshi Shinohara. How to identify a hyperbolic set as a blender. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6815-6836. doi: 10.3934/dcds.2020295

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (199)
  • HTML views (635)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy