# American Institute of Mathematical Sciences

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

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

 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  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.

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:
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##### 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
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
 $q$ 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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