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Multiple coverings of the farthest-off points with small density from projective geometry
1. | Department of Mathematics and Informatics, Perugia University, Perugia, 06123 |
2. | Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, GSP-4, Moscow, 127994, Russian Federation |
3. | Department of Mathematics and Computer Science, University of Perugia, Perugia, 06123, Italy |
References:
[1] |
N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, Small complete caps from nodal cubics, preprint, arXiv:1305.3019 |
[2] |
N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, Small complete caps from singular cubics, II, J. Algebraic Combin., 41 (2015), 185-216.
doi: 10.1007/s10801-014-0532-7. |
[3] |
U. Bartocci, Dense $k$-systems in Galois planes, Boll. Un. Mat. Ital. D(6), 2 (1983), 71-77. |
[4] |
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. |
[5] |
L. A. Bassalygo, G. V. Zaitsev and V. A. Zinov'ev, Uniformly Packed Codes, Probl. Inf. Transmis., 10 (1974), 6-9. |
[6] |
L. M. Batten and J. M. Dover, Some sets of type $(m,n)$ in cubic order planes, Des. Codes Cryptogr., 16 (1999) 211-213.
doi: 10.1023/A:1008397209409. |
[7] |
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. |
[8] |
R. A. Brualdi, V. S. Pless and R. M. Wilson, Short codes with a given covering radius, IEEE Trans. Inform. Theory, 35 (1989), 99-109.
doi: 10.1109/18.42181. |
[9] |
R. Calderbank, On uniformly packed [n,n-k,4] codes over GF(q) and a class of caps in PG(k-1,q), J. London Math. Soc., 26 (1982), 365-384.
doi: 10.1112/jlms/s2-26.2.365. |
[10] |
R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.
doi: 10.1112/blms/18.2.97. |
[11] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. |
[12] |
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. |
[13] |
A. A. Davydov, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, On constructions and parameters of symmetric configurations $v_k$, Des. Codes Cryptogr., |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
S. Fanali and M. Giulietti, On the number of rational points of generalized Fermat curves over finite fields, Int. J. Number Theory, 8 (2012), 1087-1097.
doi: 10.1142/S1793042112500650. |
[18] |
A. Gács and T. Szőnyi, Random constructions and density results, Des. Codes Cryptogr., 47 (2008), 267-287.
doi: 10.1007/s10623-007-9149-3. |
[19] |
M. Giulietti, On small dense sets in Galois planes, Electr. J. Combin., 14 (2007), #75. |
[20] |
M. Giulietti, Small complete caps in PG(N,q), q even, J. Combin. Des., 15 (2007), 420-436.
doi: 10.1002/jcd.20131. |
[21] |
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. |
[22] |
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. |
[23] |
M. Giulietti and F. Torres, On dense sets related to plane algebraic curves, Ars Combinatoria, 72 (2004), 33-40. |
[24] |
J. M. Goetals and H. C. Tilborg, Uniformly packed codes, Philips Res. Repts., 30 (1975), 9-36. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
N. Hamilton and T. Penttila, Sets of type (a,b) from subgroups of L(1,$p^R$), J. Algebr. Combin., 13 (2001), 67-76.
doi: 10.1023/A:1008775818040. |
[30] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, $2^{nd}$ edition, Oxford University Press, Oxford, 1998. |
[31] |
I. S. Honkala, On the normality of multiple covering codes, Discrete Math., 125 (1994), 229-239.
doi: 10.1016/0012-365X(94)90164-3. |
[32] |
I. Honkala and S. Litsyn, Generalizations of the covering radius problem in coding theory, Bull. Inst. Combin., 17 (1996), 39-46. |
[33] |
S. J. Kovács, Small saturated sets in finite projective planes, Rend. Mat. Ser. VII, 12 (1992), 157-164. |
[34] |
R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Reading, 1983. |
[35] |
J. H. van Lint, Codes, in Handbook of Combinatorics (eds. R. Graham, M. Grötshel, L. Lovász), MIT Press, Cambridge, MA, 1995, 772-807. |
[36] |
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. |
[37] |
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. |
[38] |
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. |
[39] |
J. Quistorff, On Codes with given minimum distance and covering radius, Beiträge Algebra Geom., 41 (2000) 469-478. |
[40] |
J. Quistorff, Correction: On codes with given minimum distance and covering radius, Beiträge Algebra Geom., 42 (2001), 601-611. |
[41] |
T. Szőnyi, Complete arcs in finite projective geometries, Ph.D thesis, Univ. L. Eötvös, Budapest, 1984. |
[42] |
T. Szőnyi, Complete arcs in Galois planes: a survey, in Quaderni del Seminario di Geometrie Combinatorie 94, Università degli Studi di Roma "La Sapienza'', Roma, 1989. |
[43] |
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] |
N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, Small complete caps from nodal cubics, preprint, arXiv:1305.3019 |
[2] |
N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, Small complete caps from singular cubics, II, J. Algebraic Combin., 41 (2015), 185-216.
doi: 10.1007/s10801-014-0532-7. |
[3] |
U. Bartocci, Dense $k$-systems in Galois planes, Boll. Un. Mat. Ital. D(6), 2 (1983), 71-77. |
[4] |
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. |
[5] |
L. A. Bassalygo, G. V. Zaitsev and V. A. Zinov'ev, Uniformly Packed Codes, Probl. Inf. Transmis., 10 (1974), 6-9. |
[6] |
L. M. Batten and J. M. Dover, Some sets of type $(m,n)$ in cubic order planes, Des. Codes Cryptogr., 16 (1999) 211-213.
doi: 10.1023/A:1008397209409. |
[7] |
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. |
[8] |
R. A. Brualdi, V. S. Pless and R. M. Wilson, Short codes with a given covering radius, IEEE Trans. Inform. Theory, 35 (1989), 99-109.
doi: 10.1109/18.42181. |
[9] |
R. Calderbank, On uniformly packed [n,n-k,4] codes over GF(q) and a class of caps in PG(k-1,q), J. London Math. Soc., 26 (1982), 365-384.
doi: 10.1112/jlms/s2-26.2.365. |
[10] |
R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.
doi: 10.1112/blms/18.2.97. |
[11] |
G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. |
[12] |
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. |
[13] |
A. A. Davydov, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, On constructions and parameters of symmetric configurations $v_k$, Des. Codes Cryptogr., |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
S. Fanali and M. Giulietti, On the number of rational points of generalized Fermat curves over finite fields, Int. J. Number Theory, 8 (2012), 1087-1097.
doi: 10.1142/S1793042112500650. |
[18] |
A. Gács and T. Szőnyi, Random constructions and density results, Des. Codes Cryptogr., 47 (2008), 267-287.
doi: 10.1007/s10623-007-9149-3. |
[19] |
M. Giulietti, On small dense sets in Galois planes, Electr. J. Combin., 14 (2007), #75. |
[20] |
M. Giulietti, Small complete caps in PG(N,q), q even, J. Combin. Des., 15 (2007), 420-436.
doi: 10.1002/jcd.20131. |
[21] |
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. |
[22] |
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. |
[23] |
M. Giulietti and F. Torres, On dense sets related to plane algebraic curves, Ars Combinatoria, 72 (2004), 33-40. |
[24] |
J. M. Goetals and H. C. Tilborg, Uniformly packed codes, Philips Res. Repts., 30 (1975), 9-36. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
N. Hamilton and T. Penttila, Sets of type (a,b) from subgroups of L(1,$p^R$), J. Algebr. Combin., 13 (2001), 67-76.
doi: 10.1023/A:1008775818040. |
[30] |
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, $2^{nd}$ edition, Oxford University Press, Oxford, 1998. |
[31] |
I. S. Honkala, On the normality of multiple covering codes, Discrete Math., 125 (1994), 229-239.
doi: 10.1016/0012-365X(94)90164-3. |
[32] |
I. Honkala and S. Litsyn, Generalizations of the covering radius problem in coding theory, Bull. Inst. Combin., 17 (1996), 39-46. |
[33] |
S. J. Kovács, Small saturated sets in finite projective planes, Rend. Mat. Ser. VII, 12 (1992), 157-164. |
[34] |
R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Reading, 1983. |
[35] |
J. H. van Lint, Codes, in Handbook of Combinatorics (eds. R. Graham, M. Grötshel, L. Lovász), MIT Press, Cambridge, MA, 1995, 772-807. |
[36] |
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. |
[37] |
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. |
[38] |
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. |
[39] |
J. Quistorff, On Codes with given minimum distance and covering radius, Beiträge Algebra Geom., 41 (2000) 469-478. |
[40] |
J. Quistorff, Correction: On codes with given minimum distance and covering radius, Beiträge Algebra Geom., 42 (2001), 601-611. |
[41] |
T. Szőnyi, Complete arcs in finite projective geometries, Ph.D thesis, Univ. L. Eötvös, Budapest, 1984. |
[42] |
T. Szőnyi, Complete arcs in Galois planes: a survey, in Quaderni del Seminario di Geometrie Combinatorie 94, Università degli Studi di Roma "La Sapienza'', Roma, 1989. |
[43] |
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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