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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.
[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.

[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.

[4]

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

[5]

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

[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. 

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

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.
[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.

[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.

[4]

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

[5]

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

[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. 

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

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
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$ 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
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