On q-ary linear completely regular codes with \rho=2 and antipodal dual

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November 2010, 4(4): 567-578. doi: 10.3934/amc.2010.4.567

On $q$-ary linear completely regular codes with $\rho=2$ and antipodal dual

Joaquim Borges ^1, , Josep Rifà ^1, and Victor A. Zinoviev ^2,

1.	Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain, Spain
2.	Institute for Problems of Information Transmission, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russian Federation

Received February 2010 Published November 2010

Abstract
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We characterize all $q$-ary linear completely regular codes with covering radius $\rho=2$ when the dual codes are antipodal. These completely regular codes are extensions of linear completely regular codes with covering radius 1, which we also classify. For $\rho=2$, we give a list of all such codes known to us. This also gives the characterization of two weight linear antipodal codes. Finally, for a class of completely regular codes with covering radius $\rho=2$ and antipodal dual, some interesting properties on self-duality and lifted codes are pointed out.

Keywords: covering radius., Linear completely regular codes, completely transitive codes.

Mathematics Subject Classification: Primary: 94B25; Secondary: 94B6.

Citation: Joaquim Borges, Josep Rifà, Victor A. Zinoviev. On $q$-ary linear completely regular codes with $\rho=2$ and antipodal dual. Advances in Mathematics of Communications, 2010, 4 (4) : 567-578. doi: 10.3934/amc.2010.4.567

References:

[1]	L. A. Bassalygo, G. V. Zaitsev and V. A. Zinoviev, Uniformly close-packed codes, Problems Inform. Transmiss., 10 (1974), 9-14.
[2]	L. A. Bassalygo and V. A. Zinoviev, A remark on uniformly packed codes, Problems Inform. Transmiss., 13 (1977), 22-25.
[3]	G. Bogdanova, V. A. Zinoviev and T. J. Todorov, On construction of $q$-ary equidistant codes, Problems Inform. Transmiss., 43 (2007), 13-36. doi: 10.1134/S0032946007040023.
[4]	A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.
[5]	J. Borges and J. Rifà, On the nonexistence of completely transitive codes, IEEE Trans. Inform. Theory, 46 (2000), 279-280. doi: 10.1109/18.817528.
[6]	J. Borges, J. Rifà and V. A. Zinoviev, Nonexistence of completely transitive codes with error-correcting capability $e > 3$, IEEE Trans. Inform. Theory, 47 (2001), 1619-1621. doi: 10.1109/18.923747.
[7]	J. Borges, J. Rifà and V. A. Zinoviev, On non-antipodal binary completely regular codes, Discrete Math., 308 (2008), 3508-3525. doi: 10.1016/j.disc.2007.07.008.
[8]	J. Borges, J. Rifà and V. A. Zinoviev, On linear completely regular codes with covering radius $\rho=1$, preprint, arXiv:0906.0550v1
[9]	A. E. Brouwer, A. M. Cohen and A. Neumaier, "Distance-Regular Graphs," Springer-Verlag, Berlin, 1989.
[10]	K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426-434. doi: 10.1214/aoms/1177729387.
[11]	A. 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.
[12]	C. J. Colbourn and J. H. Dinitz, "The CRC Handbook of Combinatorial Designs," CRC Press, Boca Raton, FL, 1996. doi: 10.1201/9781420049954.
[13]	G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, "Covering Codes," Elsevier Science, The Nederlands, 1997.
[14]	P. Delsarte, Two-weight linear codes and strongly regular graphs, MBLE Research Laboratory, Report R160, 1971.
[15]	P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, 10 (1973), vi+97.
[16]	D. G. Fon-Der-Flaas, Perfect $2$-coloring of hypercube, Siberian Math. J., 48 (2007), 923-930. doi: 10.1007/s11202-007-0075-4.
[17]	D. G. Fon-Der-Flaas, Perfect $2$-coloring of the $12$-cube that attain the bound on correlation immunity, Siberian Electronic Math. Reports, 4 (2007), 292-295.
[18]	M. Giudici and C. E. Praeger, Completely transitive codes in Hamming graphs, Europ. J. Combinatorics, 20 (1999), 647-662. doi: 10.1006/eujc.1999.0313.
[19]	J. M. Goethals and H. C. A. Van Tilborg, Uniformly packed codes, Philips Res., 30 (1975), 9-36.
[20]	J. H. Koolen, W. S. Lee and W. J. Martin, Arithmetic completely regular codes, preprint, arXiv:0911.1826v1
[21]	F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Techn. J., 42 (1963), 79-84.
[22]	F. J. MacWilliams and N. J. A. Sloane, "The Theory if Error-Correcting Codes," Elsevier, North-Holland, 1977.
[23]	A. Neumaier, Completely regular codes, Discrete Math., 106/107 (1992), 353-360. doi: 10.1016/0012-365X(92)90565-W.
[24]	J. Rifà and V. A. Zinoviev, On new completely regular $q$-ary codes, Problems Inform. Transmiss., 43 (2007), 97-112. doi: 10.1134/S0032946007020032.
[25]	J. Rifà and V. A. Zinoviev, New completely regular $q$-ary codes, based on Kronecker products, IEEE Trans. Inform. Theory, 56 (2010), 266-272. doi: 10.1109/TIT.2009.2034812.
[26]	J. Rifà and V. A. Zinoviev, On lifting perfect codes, preprint, arXiv:1002.0295
[27]	N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Class of maximal equidistant codes, Problems Inform. Transmiss., 5 (1969), 84-87.
[28]	N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Uniformly close-packed codes, Problems Inform. Transmiss., 7 (1971), 38-50.
[29]	J. Singer, A theorem in finite projective geometry, and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.
[30]	P. Solé, Completely regular codes and completely transitive codes, Discrete Math., 81 (1990), 193-201. doi: 10.1016/0012-365X(90)90152-8.
[31]	A. Tietäväinen, On the non-existence of perfect codes over finite fields, SIAM J. Appl. Math., 24 (1973), 88-96. doi: 10.1137/0124010.
[32]	H. C. A. Van Tilborg, "Uniformly Packed Codes," Ph.D thesis, Eindhoven Univ. of Tech., 1976.
[33]	V. A. Zinoviev and V. K. Leontiev, The nonexistence of perfect codes over Galois fields, Problems Control Inform. Th., 2 (1973), 16-24.

show all references

References:

[1]	L. A. Bassalygo, G. V. Zaitsev and V. A. Zinoviev, Uniformly close-packed codes, Problems Inform. Transmiss., 10 (1974), 9-14.
[2]	L. A. Bassalygo and V. A. Zinoviev, A remark on uniformly packed codes, Problems Inform. Transmiss., 13 (1977), 22-25.
[3]	G. Bogdanova, V. A. Zinoviev and T. J. Todorov, On construction of $q$-ary equidistant codes, Problems Inform. Transmiss., 43 (2007), 13-36. doi: 10.1134/S0032946007040023.
[4]	A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.
[5]	J. Borges and J. Rifà, On the nonexistence of completely transitive codes, IEEE Trans. Inform. Theory, 46 (2000), 279-280. doi: 10.1109/18.817528.
[6]	J. Borges, J. Rifà and V. A. Zinoviev, Nonexistence of completely transitive codes with error-correcting capability $e > 3$, IEEE Trans. Inform. Theory, 47 (2001), 1619-1621. doi: 10.1109/18.923747.
[7]	J. Borges, J. Rifà and V. A. Zinoviev, On non-antipodal binary completely regular codes, Discrete Math., 308 (2008), 3508-3525. doi: 10.1016/j.disc.2007.07.008.
[8]	J. Borges, J. Rifà and V. A. Zinoviev, On linear completely regular codes with covering radius $\rho=1$, preprint, arXiv:0906.0550v1
[9]	A. E. Brouwer, A. M. Cohen and A. Neumaier, "Distance-Regular Graphs," Springer-Verlag, Berlin, 1989.
[10]	K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426-434. doi: 10.1214/aoms/1177729387.
[11]	A. 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.
[12]	C. J. Colbourn and J. H. Dinitz, "The CRC Handbook of Combinatorial Designs," CRC Press, Boca Raton, FL, 1996. doi: 10.1201/9781420049954.
[13]	G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, "Covering Codes," Elsevier Science, The Nederlands, 1997.
[14]	P. Delsarte, Two-weight linear codes and strongly regular graphs, MBLE Research Laboratory, Report R160, 1971.
[15]	P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, 10 (1973), vi+97.
[16]	D. G. Fon-Der-Flaas, Perfect $2$-coloring of hypercube, Siberian Math. J., 48 (2007), 923-930. doi: 10.1007/s11202-007-0075-4.
[17]	D. G. Fon-Der-Flaas, Perfect $2$-coloring of the $12$-cube that attain the bound on correlation immunity, Siberian Electronic Math. Reports, 4 (2007), 292-295.
[18]	M. Giudici and C. E. Praeger, Completely transitive codes in Hamming graphs, Europ. J. Combinatorics, 20 (1999), 647-662. doi: 10.1006/eujc.1999.0313.
[19]	J. M. Goethals and H. C. A. Van Tilborg, Uniformly packed codes, Philips Res., 30 (1975), 9-36.
[20]	J. H. Koolen, W. S. Lee and W. J. Martin, Arithmetic completely regular codes, preprint, arXiv:0911.1826v1
[21]	F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Techn. J., 42 (1963), 79-84.
[22]	F. J. MacWilliams and N. J. A. Sloane, "The Theory if Error-Correcting Codes," Elsevier, North-Holland, 1977.
[23]	A. Neumaier, Completely regular codes, Discrete Math., 106/107 (1992), 353-360. doi: 10.1016/0012-365X(92)90565-W.
[24]	J. Rifà and V. A. Zinoviev, On new completely regular $q$-ary codes, Problems Inform. Transmiss., 43 (2007), 97-112. doi: 10.1134/S0032946007020032.
[25]	J. Rifà and V. A. Zinoviev, New completely regular $q$-ary codes, based on Kronecker products, IEEE Trans. Inform. Theory, 56 (2010), 266-272. doi: 10.1109/TIT.2009.2034812.
[26]	J. Rifà and V. A. Zinoviev, On lifting perfect codes, preprint, arXiv:1002.0295
[27]	N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Class of maximal equidistant codes, Problems Inform. Transmiss., 5 (1969), 84-87.
[28]	N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Uniformly close-packed codes, Problems Inform. Transmiss., 7 (1971), 38-50.
[29]	J. Singer, A theorem in finite projective geometry, and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.
[30]	P. Solé, Completely regular codes and completely transitive codes, Discrete Math., 81 (1990), 193-201. doi: 10.1016/0012-365X(90)90152-8.
[31]	A. Tietäväinen, On the non-existence of perfect codes over finite fields, SIAM J. Appl. Math., 24 (1973), 88-96. doi: 10.1137/0124010.
[32]	H. C. A. Van Tilborg, "Uniformly Packed Codes," Ph.D thesis, Eindhoven Univ. of Tech., 1976.
[33]	V. A. Zinoviev and V. K. Leontiev, The nonexistence of perfect codes over Galois fields, Problems Control Inform. Th., 2 (1973), 16-24.

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