# American Institute of Mathematical Sciences

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, 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
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
 [1] 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 [2] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [3] Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071 [4] Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102 [5] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [6] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 [7] Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162 [8] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [9] Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045 [10] Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 [11] Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 [12] Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 [13] Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $\beta$-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 [14] Duy Phan. Approximate controllability for Navier–Stokes equations in $\rm3D$ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062 [15] Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094 [16] Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117 [17] Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106 [18] Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046 [19] Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127 [20] Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015

2019 Impact Factor: 0.734

## Tools

Article outline

Figures and Tables