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Small weight codewords in the codes arising from Desarguesian projective planes

Veerle Fack, Szabolcs L. Fancsali, L. Storme, Geetrui Van de Voorde, Joost Winne
2007 Designs, Codes and Cryptography  
We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K.  ...  Chouinard on codewords of small weight in the codes arising from P G(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes.  ...  For arbitrary Desarguesian projective planes, we give a discrete spectrum for the possible weights of the codewords in the interval [q + 2, 3q/2] and exclude all codewords with weight in the interval [  ... 
doi:10.1007/s10623-007-9126-x fatcat:4p5jprjpp5hffg475wqpquo5im

On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

Michel Lavrauw, Leo Storme, Geertrui Van de Voorde
2008 Designs, Codes and Cryptography  
We link the codewords of small weight of this code to blocking sets with respect to lines in P G(n, q) and we exclude all possible codewords arising from small linear blocking sets.  ...  In this paper, we study the p-ary linear code C(P G(n, q)), q = p h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space P G(n, q), and its dual  ...  Acknowledgement: The authors wish to thank the referees for their detailed reading of this article and for their valuable suggestions for improvement.  ... 
doi:10.1007/s10623-008-9203-9 fatcat:qkjipzxtrrc5hhpa5zddv5oyv4

On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual [article]

Michel Lavrauw, Leo Storme, Geertrui Van de Voorde
2012 arXiv   pre-print
We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n, q) and we exclude all possible codewords arising from small linear blocking sets.  ...  In this paper, we study the p-ary linear code C(PG(n, q)), q = p^h, p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual  ...  Acknowledgement: The authors wish to thank the referees for their detailed reading of this article and for their valuable suggestions for improvement.  ... 
arXiv:1201.3290v1 fatcat:nhd2z3biwzawdasnzz4st3d6qe

An upper bound for the minimum weight of the dual codes of desarguesian planes

J.D. Key, T.P. McDonough, V.C. Mavron
2009 European journal of combinatorics (Print)  
We show that a construction described in Clark, Key and de Resmini [9] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian  ...  This gives words of weight 2p m + 1 − p m −1 p−1 in the dual of the p-ary code of the desarguesian plane of order p m , and provides an improved upper bound for the minimum weight of the dual code.  ...  For planes of small non-prime odd order, the dual ternary codes of the four projective planes of order 9 were shown in [16] to all have d ⊥ = 15 apart from the Hughes plane, for which d ⊥ = 14.  ... 
doi:10.1016/j.ejc.2008.01.003 fatcat:mflozklhfzbdnjtxodp5iutet4

Moderate Density Parity-Check Codes from Projective Bundles [article]

Jessica Bariffi, Sam Mattheus, Alessandro Neri, Joachim Rosenthal
2021 arXiv   pre-print
We design a parity-check matrix for this family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence  ...  In this setting, our codes have the best possible error-correction performance for this range of parameters.  ...  Acknowledgment The work of A. Neri was supported by the Swiss National Science Foundation through grant no. 187711. The work of J.  ... 
arXiv:2103.09722v1 fatcat:tsvuul4l3zedfggz5klvs6pj2m

Small weight codewords in the LDPC codes arising from linear representations of geometries

V. Pepe, L. Storme, G. Van de Voorde
2009 Journal of combinatorial designs (Print)  
In this paper, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods.  ...  When investigating the small weight codewords of T * 2 (Θ) D , we deal with the case of Θ a regular hyperoval, i.e. a conic and its nucleus, separately, since in this case, we have a larger upper bound  ...  The goal of this section is to characterize small weight codewords of the q-ary LDPC code C arising from T * 2 (Θ), with q = 2 h , by using geometrical arguments.  ... 
doi:10.1002/jcd.20179 fatcat:h7bc3r2p7ben7fgz4qu7sj7rwq

Scattered Spaces in Galois Geometry [article]

Michel Lavrauw
2016 arXiv   pre-print
This is a survey paper on the theory of scattered spaces in Galois geometry and its applications.  ...  The author thanks the members of the Algebra group of Sabancı University for their hospitality and the anonymous referees for their comments and suggestions.  ...  For non-Desarguesian projective planes, and in particular for translation planes, the classification remains open.  ... 
arXiv:1512.05251v2 fatcat:xhhabrelpzazvi45fjgvprcbt4

Partitions in finite geometry and related constant composition codes

Timothy Alderson, Keith Mellinger
2008 Innovations in Incidence Geometry Algebraic Topological and Combinatorial  
We look at the construction of constant composition codes (CCCs) from various types of partitions in finite projective spaces.  ...  In addition, we examine other partitions of objects embedded in finite projective spaces and their associated codes.  ...  The authors would like to thank Norm Johnson for his guidance regarding the proof of Theorem 2.6.  ... 
doi:10.2140/iig.2008.8.49 fatcat:q7emi2h775fj7b3zv5n2qg373q

On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual [article]

Michel Lavrauw, Leo Storme, Geertrui Van de Voorde
2012 arXiv   pre-print
We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets.  ...  Next, we study the dual code of Ck(n, q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [12] to general dimension.  ...  the codewords of small weight in the p-ary linear code generated by the incidence matrix of points and k-dimensional spaces, or for short k-spaces, in P G(n, q), q = p h , p prime, h ≥ 1.  ... 
arXiv:1201.3291v1 fatcat:6lo5viimajboxidjrbzkawqyp4

On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual

M. Lavrauw, L. Storme, G. Van de Voorde
2008 Finite Fields and Their Applications  
We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets.  ...  Next, we study the dual code of C k (n, q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H.  ...  the codewords of small weight in the p-ary linear code generated by the incidence matrix of points and k-dimensional spaces, or for short k-spaces, in PG(n, q), q = p h , p prime, h 1.  ... 
doi:10.1016/j.ffa.2008.06.002 fatcat:hpifrjnznna5nmdvv5fyle3wcq

Small sets of even type and codewords

J. D. Key, M. J. de Resmini
1998 Journal of Geometry  
In particular we look at small sets of points in projective planes of even order that are met evenly by all the lines of the plane, and find vectors of small weight in the binary hull and in the code's  ...  We examine some geometric configurations of points in designs that give rise to vectors in the codes associated with the designs.  ...  and the first author would like to thank the Dipartimento di Matematica at the Universitá di Roma 'La Sapienza' for their hospitality, and the C.N.R.  ... 
doi:10.1007/bf01237498 fatcat:nfxfdrwpn5a7ndpvd3ynkylyye

Some Applications of Magma in Designs and Codes: Oval Designs, Hermitian Unitals and Generalized Reed–Muller Codes

J.D. Key
2001 Journal of symbolic computation  
We describe three applications of Magma to problems in the area of designs and the associated codes: • Steiner systems, Hadamard designs and symmetric designs arising from a oval in an even order plane  ...  , leading in the classical case to bent functions and difference-set designs; • the hermitian unital as a 2-(q 3 + 1, q + 1, 1) design, and the code over F p where p divides q + 1; • a basis of minimum-weight  ...  Acknowledgement: The author would like to thank the Department of Computer Science and Engineering and the Center for Communication and Information Science (CCIS) at the University of Nebraska for their  ... 
doi:10.1006/jsco.1999.1007 fatcat:4x6vyvqhq5dk3jpxi2mfzkf3qy

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

Crnković Dean, ,Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia, Mostarac Nina, Rodrigues Bernardo G., Storme Leo, ,Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0002, Pretoria, South Africa, ,Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Ghent, Belgium
2019 Advances in Mathematics of Communications  
In this paper we construct 2-PD-sets of 16 elements for codes from the Desarguesian projective planes PG(2, q), where q = 2 h and 5 ≤ h ≤ 9.  ...  We also construct 3-PD-sets of 75 elements for the code from the Desarguesian projective plane PG(2, q), where q = 2 9 .  ...  The research of D. Crnković and N. Mostarac has been supported by the Croatian Science Foundation under the project 6732.  ... 
doi:10.3934/amc.2020075 fatcat:32nhzjcczffwhoyo5elhhfc37y

On the history of ring geometry (with a thematical overview of literature) [article]

Dirk Keppens
2020 arXiv   pre-print
In this survey paper we give an historical and at the same time thematical overview of the development of ring geometry from its origin to the current state of the art.  ...  A comprehensive up-to-date list of literature is added with articles that treat ring geometry within the scope of incidence geometry.  ...  Another aspect that appears in the literature is the correspondence between two-weight codes and strongly regular graphs.  ... 
arXiv:2003.02881v1 fatcat:ad7sidz75bgfxcd4sw3yftb45m

The Use of Blocking Sets in Galois Geometries and in Related Research Areas [chapter]

V. Pepe, L. Storme
2011 Springer Proceedings in Mathematics  
Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research  ...  By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems  ...  One might wonder where all these Baer subplanes of holes in these poor planes arise from.  ... 
doi:10.1007/978-1-4614-0709-6_15 fatcat:ngfvmymvprbqfooj4do3jadnb4
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