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The $[46, 9, 20]_2$ code is unique
$ 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 |
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.
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. |
Code | |||||
[1057,244, 33] | 16 | 813 | 180 | 3 | |
[4161,730, 65] | 32 | 3431 | 1623 | 3 | |
[16513, 2188,129] | 64 | 14325 | 40696 | 3 | |
[65793, 6562,257] | 128 | 59231 | 3965945 | 3 | |
[262657, 19684,513] | 256 | 242973 | 3625171287 | 3 |
Code | |||||
[1057,244, 33] | 16 | 813 | 180 | 3 | |
[4161,730, 65] | 32 | 3431 | 1623 | 3 | |
[16513, 2188,129] | 64 | 14325 | 40696 | 3 | |
[65793, 6562,257] | 128 | 59231 | 3965945 | 3 | |
[262657, 19684,513] | 256 | 242973 | 3625171287 | 3 |
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