Multiple coverings of the farthest-off points with small density from projective geometry

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February 2015, 9(1): 63-85. doi: 10.3934/amc.2015.9.63

Multiple coverings of the farthest-off points with small density from projective geometry

Daniele Bartoli ^1, , Alexander A. Davydov ^2, , Massimo Giulietti ^3, , Stefano Marcugini ^1, and Fernanda Pambianco ^1,

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

Received May 2014 Published February 2015

Abstract
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In this paper we deal with the special class of covering codes consisting of multiple coverings of the farthest-off points (MCF). In order to measure the quality of an MCF code, we use a natural extension of the notion of density for ordinary covering codes, that is the $\mu$-density for MCF codes; a generalization of the length function for linear covering codes is also introduced. Our main results consist in a number of upper bounds on such a length function, obtained through explicit constructions, especially for the case of covering radius $R=2$. A key tool is the possibility of computing the $\mu$-length function in terms of Projective Geometry over finite fields. In fact, linear $(R,\mu )$-MCF codes with parameters $ [n,n-r,d]_{q}R$ have a geometrical counterpart consisting of special subsets of $n$ points in the projective space $PG(n-r-1,q)$. We introduce such objects under the name of $(\rho,\mu)$-saturating sets and we provide a number of example and existence results. Finally, Almost Perfect MCF (APMCF) codes, that is codes for which each word at distance $R$ from the code belongs to {exactly} $\mu $ spheres centered in codewords, are considered and their connections with uniformly packed codes, two-weight codes, and subgroups of Singer groups are pointed out.

Keywords: saturating sets in projective spaces, Multiple coverings, covering density., covering codes, farthest-off points.

Mathematics Subject Classification: Primary: 51E21, 51E22; Secondary: 94B0.

Citation: Daniele Bartoli, Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Multiple coverings of the farthest-off points with small density from projective geometry. Advances in Mathematics of Communications, 2015, 9 (1) : 63-85. doi: 10.3934/amc.2015.9.63

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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