\(s\)-PD-sets for codes from projective planes \(\mathrm{PG}(2,2^h)\), \(5\leq h\leq 9\). (English) Zbl 1471.51004
MacWilliams in 1964 introduced the permutation decoding, an algorithm to decode error-correcting codes. Such algorithm uses PD-sets, that is sets of the code automorphism defined with respect to a given information set of the code. Not all codes can be decoded using this method, it will depend on whether or not their automorphism codes contain PD-sets.
More precisely, consider a linear code \(C\subseteq \mathbb{F}_p^n\) which can correct at most \(t\) errors and consider \(I\) to be an information set for \(C\). A subset \(S\) of the automorphism group of \(C\) is said to be a PD-set for \(C\) if every \(t\)-set of coordinate positions can be moved by at least one element of \(S\) out of the information set \(I\). For \(s\leq t\), an \(s\)-PD-set is a set \(S\) of the automorphism group of \(C\) such that every \(s\)-set of coordinate positions can be moved by at least one element of \(S\) out of the information set \(I\).
In this paper, the authors construct \(s\)-PD-sets with \(s \in \{2,3\}\) for the linear codes arising from the incidence matrices of points and lines of \(\mathrm{PG}(2,q)\), for \(q=2^h\) and \(h \in \{5,6,7,8,9\}\). They use a special basis for these codes which was found by P. Vandendriessche [Finite Fields Appl. 17, No. 6, 521–531 (2011; Zbl 1248.05026)].
More precisely, consider a linear code \(C\subseteq \mathbb{F}_p^n\) which can correct at most \(t\) errors and consider \(I\) to be an information set for \(C\). A subset \(S\) of the automorphism group of \(C\) is said to be a PD-set for \(C\) if every \(t\)-set of coordinate positions can be moved by at least one element of \(S\) out of the information set \(I\). For \(s\leq t\), an \(s\)-PD-set is a set \(S\) of the automorphism group of \(C\) such that every \(s\)-set of coordinate positions can be moved by at least one element of \(S\) out of the information set \(I\).
In this paper, the authors construct \(s\)-PD-sets with \(s \in \{2,3\}\) for the linear codes arising from the incidence matrices of points and lines of \(\mathrm{PG}(2,q)\), for \(q=2^h\) and \(h \in \{5,6,7,8,9\}\). They use a special basis for these codes which was found by P. Vandendriessche [Finite Fields Appl. 17, No. 6, 521–531 (2011; Zbl 1248.05026)].
Reviewer: Ferdinando Zullo (Caserta)
MSC:
| 51E22 | Linear codes and caps in Galois spaces |
| 51E20 | Combinatorial structures in finite projective spaces |
| 51A30 | Desarguesian and Pappian geometries |
Citations:
Zbl 1248.05026References:
| [1] | D. Crnković; N. Mostarac, PD-sets for codes related to flag-transitive symmetric designs, Trans. Comb., 7, 37-50 (2018) · Zbl 1463.05026 · doi:10.22108/toc.2017.21615 |
| [2] | D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28, 541-543 (1982) · Zbl 0479.94021 · doi:10.1109/TIT.1982.1056504 |
| [3] | J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2^nd edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. · Zbl 0899.51002 |
| [4] | W. C. Huffman, Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2, 1345-1440 (1998) · Zbl 0926.94039 |
| [5] | 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, 172-201 (2011) · Zbl 1366.94705 |
| [6] | J. D. Key; T. P. McDonough; V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26, 665-682 (2005) · Zbl 1074.94014 · doi:10.1016/j.ejc.2004.04.007 |
| [7] | J. MacWilliams, Permutation decoding of systematic codes, Bell Syst. Tech. J., 43, 485-505 (1964) · Zbl 0116.35304 · doi:10.1002/j.1538-7305.1964.tb04075.x |
| [8] | 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. · Zbl 0369.94008 |
| [9] | G. E. Moorhouse, Bruck nets, codes, and characters of loops, Des. Codes Cryptogr., 1, 7-29 (1991) · Zbl 0755.05020 · doi:10.1007/BF00123956 |
| [10] | N. Pace; A. Sonnino, On linear codes admitting large automorphism groups, Des. Codes Cryptogr., 83, 115-143 (2017) · Zbl 1368.51011 · doi:10.1007/s10623-016-0207-6 |
| [11] | 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, 122-129 (1969) · Zbl 0185.24305 · doi:10.1016/S0021-9800(69)80046-3 |
| [12] | 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, 521-531 (2011) · Zbl 1248.05026 · doi:10.1016/j.ffa.2011.03.003 |
| [13] | 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. |
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