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Further results on multiple coverings of the farthest-off points
1. | Department of Mathematics, Ghent University, Gent, 9000, Belgium |
2. | Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, GSP-4, Moscow, 127994, Russian Federation |
3. | Department of Mathematics and Informatics, Perugia University, Perugia, 06123, Italy, Italy, Italy |
References:
[1] |
D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Multiple coverings of the farthest-off points and multiple saturating sets in projective spaces, in Proc. XIII Int. Workshop Algebr. Combin. Coding Theory, ACCT2012, Pomoria, Bulgaria, 2012, 53-59. |
[2] |
D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Multiple coverings of the farthest-off points with small density from projective geometry, Adv. Math. Commun., 9 (2015), 63-85.
doi: 10.3934/amc.2015.9.63. |
[3] |
D. Bartoli, G. Faina, S. Marcugini and F. Pambianco, On the minimum size of complete arcs and minimal saturating sets in projective planes, J. Geom., 104 (2013) 409-419.
doi: 10.1007/s00022-013-0178-y. |
[4] |
E. Boros, T. Szőnyi and K. Tichler, On defining sets for projective planes, Discrete Math., 303 (2005), 17-31.
doi: 10.1016/j.disc.2004.12.015. |
[5] |
R. A. Brualdi, S. Litsyn and V. S. Pless, Covering radius, in Handbook of Coding Theory (eds. V.S. Pless, W.C. Huffman and R.A. Brualdi), Elsevier, Amsterdam, 1998, 755-826. |
[6] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. |
[7] |
A. A. Davydov, Constructions and families of covering codes and saturated sets of points in projective geometry, IEEE Trans. Inform. Theory, 41 (1995), 2071-2080.
doi: 10.1109/18.476339. |
[8] |
A. A. Davydov, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, On constructions and parameters of symmetric configurations $v_k$, Des. Codes Cryptogr., 80 (2016), 125-147.
doi: 10.1007/s10623-015-0070-x. |
[9] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, On the spectrum of possible parameters of symmetric configurations, in Proc. XII Int. Symp. Probl. Redundancy Inform. Control Systems, Saint-Petersburg, Russia, 2009, 59-64. |
[10] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, New inductive constructions of complete caps in $PG(N,q)$, $q$ even, J. Comb. Des., 18 (2010), 176-201. |
[11] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Linear nonbinary covering codes and saturating sets in projective spaces, Adv. Math. Commun., 5 (2011), 119-147.
doi: 10.3934/amc.2011.5.119. |
[12] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Some combinatorial aspects of constructing bipartite-graph codes, Graphs Combin., 29 (2013), 187-212.
doi: 10.1007/s00373-011-1103-5. |
[13] |
M. Giulietti, On small dense sets in Galois planes, Electr. J. Combin., 14 (2007), #75. |
[14] |
M. Giulietti, Small complete caps in $PG(N,q), q$ even, J. Combin. Des., 15 (2007), 420-436.
doi: 10.1002/jcd.20131. |
[15] |
M. Giulietti, The geometry of covering codes: small complete caps and saturating sets in Galois spaces, in Surveys in Combinatorics, Cambridge Univ. Press, 2013, 51-90. |
[16] |
M. Giulietti and F. Pasticci, Quasi-perfect linear codes with minimum distance 4, IEEE Trans. Inform. Theory, 53 (2007), 1928-1935.
doi: 10.1109/TIT.2007.894688. |
[17] |
M. Giulietti and F. Torres, On dense sets related to plane algebraic curves, Ars Combin., 72 (2004), 33-40. |
[18] |
H. O. Hämäläinen, I. S. Honkala, M. K. Kaikkonen and S. N. Litsyn, Bounds for binary multiple covering codes, Des. Codes Cryptogr., 3 (1993), 251-275.
doi: 10.1007/BF01388486. |
[19] |
H. O. Hämäläinen, I. S. Honkala, S. N. Litsyn and P. R. J. Östergård, Bounds for binary codes that are multiple coverings of the farthest-off points, SIAM J. Discrete Math., 8 (1995), 196-207.
doi: 10.1137/S0895480193252100. |
[20] |
H. Hämäläinen, I. Honkala, S. Litsyn and P. Östergård, Football pools - a game for mathematicians, Amer. Math. Monthly, 102 (1995), 579-588.
doi: 10.2307/2974552. |
[21] |
H. O. Hämäläinen and S. Rankinen, Upper bounds for football pool problems and mixed covering codes, J. Combin. Theory Ser. A, 56 (1991), 84-95.
doi: 10.1016/0097-3165(91)90024-B. |
[22] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Univ. Press, 1998. |
[23] |
I. S. Honkala, On the normality of multiple covering codes, Discrete Math., 125 (1994), 229-239.
doi: 10.1016/0012-365X(94)90164-3. |
[24] |
I. Honkala and S. Litsyn, Generalizations of the covering radius problem in coding theory, Bull. Inst. Combin., 17 (1996), 39-46. |
[25] |
P. R. J. Östergård and H. O. Hämäläinen, A new table of binary/ternary mixed covering codes, Des. Codes Cryptogr., 11 (1997), 151-178.
doi: 10.1023/A:1008228721072. |
[26] |
F. Pambianco, D. Bartoli, G. Faina and S. Marcugini, Classification of the smallest minimal 1-saturating sets in $PG(2,q)$, $q\le 23$, Electron. Notes Discrete Math., 40 (2013) 229-233. |
[27] |
F. Pambianco, A. A. Davydov, D. Bartoli, M. Giulietti and S. Marcugini, A note on multiple coverings of the farthest-off points, Electron. Notes Discrete Math., 40 (2013) 289-293. |
[28] |
J. Quistorff, On Codes with given minimum distance and covering radius, Beiträge Algebra Geom., 41 (2000) 469-478. |
[29] |
J. Quistorff, Correction: On codes with given minimum distance and covering radius, Beiträge Algebra Geom., 42 (2001), 601-611. |
[30] |
T. Szőnyi, Complete arcs in Galois planes: a survey, in Quaderni del Seminario di Geometrie Combinatorie 94, Univ. degli Studi di Roma "La Sapienza'', Roma, 1989. |
[31] |
G. J. M. van Wee, G. D. Cohen and S. N. Litsyn, A note on perfect multiple coverings of the Hamming space, IEEE Trans. Inform. Theory, 37 (1991), 678-682. |
show all references
References:
[1] |
D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Multiple coverings of the farthest-off points and multiple saturating sets in projective spaces, in Proc. XIII Int. Workshop Algebr. Combin. Coding Theory, ACCT2012, Pomoria, Bulgaria, 2012, 53-59. |
[2] |
D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Multiple coverings of the farthest-off points with small density from projective geometry, Adv. Math. Commun., 9 (2015), 63-85.
doi: 10.3934/amc.2015.9.63. |
[3] |
D. Bartoli, G. Faina, S. Marcugini and F. Pambianco, On the minimum size of complete arcs and minimal saturating sets in projective planes, J. Geom., 104 (2013) 409-419.
doi: 10.1007/s00022-013-0178-y. |
[4] |
E. Boros, T. Szőnyi and K. Tichler, On defining sets for projective planes, Discrete Math., 303 (2005), 17-31.
doi: 10.1016/j.disc.2004.12.015. |
[5] |
R. A. Brualdi, S. Litsyn and V. S. Pless, Covering radius, in Handbook of Coding Theory (eds. V.S. Pless, W.C. Huffman and R.A. Brualdi), Elsevier, Amsterdam, 1998, 755-826. |
[6] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. |
[7] |
A. A. Davydov, Constructions and families of covering codes and saturated sets of points in projective geometry, IEEE Trans. Inform. Theory, 41 (1995), 2071-2080.
doi: 10.1109/18.476339. |
[8] |
A. A. Davydov, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, On constructions and parameters of symmetric configurations $v_k$, Des. Codes Cryptogr., 80 (2016), 125-147.
doi: 10.1007/s10623-015-0070-x. |
[9] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, On the spectrum of possible parameters of symmetric configurations, in Proc. XII Int. Symp. Probl. Redundancy Inform. Control Systems, Saint-Petersburg, Russia, 2009, 59-64. |
[10] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, New inductive constructions of complete caps in $PG(N,q)$, $q$ even, J. Comb. Des., 18 (2010), 176-201. |
[11] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Linear nonbinary covering codes and saturating sets in projective spaces, Adv. Math. Commun., 5 (2011), 119-147.
doi: 10.3934/amc.2011.5.119. |
[12] |
A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Some combinatorial aspects of constructing bipartite-graph codes, Graphs Combin., 29 (2013), 187-212.
doi: 10.1007/s00373-011-1103-5. |
[13] |
M. Giulietti, On small dense sets in Galois planes, Electr. J. Combin., 14 (2007), #75. |
[14] |
M. Giulietti, Small complete caps in $PG(N,q), q$ even, J. Combin. Des., 15 (2007), 420-436.
doi: 10.1002/jcd.20131. |
[15] |
M. Giulietti, The geometry of covering codes: small complete caps and saturating sets in Galois spaces, in Surveys in Combinatorics, Cambridge Univ. Press, 2013, 51-90. |
[16] |
M. Giulietti and F. Pasticci, Quasi-perfect linear codes with minimum distance 4, IEEE Trans. Inform. Theory, 53 (2007), 1928-1935.
doi: 10.1109/TIT.2007.894688. |
[17] |
M. Giulietti and F. Torres, On dense sets related to plane algebraic curves, Ars Combin., 72 (2004), 33-40. |
[18] |
H. O. Hämäläinen, I. S. Honkala, M. K. Kaikkonen and S. N. Litsyn, Bounds for binary multiple covering codes, Des. Codes Cryptogr., 3 (1993), 251-275.
doi: 10.1007/BF01388486. |
[19] |
H. O. Hämäläinen, I. S. Honkala, S. N. Litsyn and P. R. J. Östergård, Bounds for binary codes that are multiple coverings of the farthest-off points, SIAM J. Discrete Math., 8 (1995), 196-207.
doi: 10.1137/S0895480193252100. |
[20] |
H. Hämäläinen, I. Honkala, S. Litsyn and P. Östergård, Football pools - a game for mathematicians, Amer. Math. Monthly, 102 (1995), 579-588.
doi: 10.2307/2974552. |
[21] |
H. O. Hämäläinen and S. Rankinen, Upper bounds for football pool problems and mixed covering codes, J. Combin. Theory Ser. A, 56 (1991), 84-95.
doi: 10.1016/0097-3165(91)90024-B. |
[22] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Univ. Press, 1998. |
[23] |
I. S. Honkala, On the normality of multiple covering codes, Discrete Math., 125 (1994), 229-239.
doi: 10.1016/0012-365X(94)90164-3. |
[24] |
I. Honkala and S. Litsyn, Generalizations of the covering radius problem in coding theory, Bull. Inst. Combin., 17 (1996), 39-46. |
[25] |
P. R. J. Östergård and H. O. Hämäläinen, A new table of binary/ternary mixed covering codes, Des. Codes Cryptogr., 11 (1997), 151-178.
doi: 10.1023/A:1008228721072. |
[26] |
F. Pambianco, D. Bartoli, G. Faina and S. Marcugini, Classification of the smallest minimal 1-saturating sets in $PG(2,q)$, $q\le 23$, Electron. Notes Discrete Math., 40 (2013) 229-233. |
[27] |
F. Pambianco, A. A. Davydov, D. Bartoli, M. Giulietti and S. Marcugini, A note on multiple coverings of the farthest-off points, Electron. Notes Discrete Math., 40 (2013) 289-293. |
[28] |
J. Quistorff, On Codes with given minimum distance and covering radius, Beiträge Algebra Geom., 41 (2000) 469-478. |
[29] |
J. Quistorff, Correction: On codes with given minimum distance and covering radius, Beiträge Algebra Geom., 42 (2001), 601-611. |
[30] |
T. Szőnyi, Complete arcs in Galois planes: a survey, in Quaderni del Seminario di Geometrie Combinatorie 94, Univ. degli Studi di Roma "La Sapienza'', Roma, 1989. |
[31] |
G. J. M. van Wee, G. D. Cohen and S. N. Litsyn, A note on perfect multiple coverings of the Hamming space, IEEE Trans. Inform. Theory, 37 (1991), 678-682. |
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Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
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