American Institute of Mathematical Sciences

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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

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.  Google Scholar

[2]

L. A. Bassalygo and V. A. Zinoviev, A remark on uniformly packed codes, Problems Inform. Transmiss., 13 (1977), 22-25.  Google Scholar

[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.  Google Scholar

[4]

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Borges, J. Rifà and V. A. Zinoviev, On linear completely regular codes with covering radius $\rho=1$,, preprint, ().   Google Scholar

[9]

A. E. Brouwer, A. M. Cohen and A. Neumaier, "Distance-Regular Graphs," Springer-Verlag, Berlin, 1989.  Google Scholar

[10]

K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426-434. doi: 10.1214/aoms/1177729387.  Google Scholar

[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.  Google Scholar

[12]

C. J. Colbourn and J. H. Dinitz, "The CRC Handbook of Combinatorial Designs," CRC Press, Boca Raton, FL, 1996. doi: 10.1201/9781420049954.  Google Scholar

[13]

G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, "Covering Codes," Elsevier Science, The Nederlands, 1997.  Google Scholar

[14]

P. Delsarte, Two-weight linear codes and strongly regular graphs, MBLE Research Laboratory, Report R160, 1971. Google Scholar

[15]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, 10 (1973), vi+97.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. M. Goethals and H. C. A. Van Tilborg, Uniformly packed codes, Philips Res., 30 (1975), 9-36.  Google Scholar

[20]

J. H. Koolen, W. S. Lee and W. J. Martin, Arithmetic completely regular codes,, preprint, ().   Google Scholar

[21]

F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Techn. J., 42 (1963), 79-84.  Google Scholar

[22]

F. J. MacWilliams and N. J. A. Sloane, "The Theory if Error-Correcting Codes," Elsevier, North-Holland, 1977.  Google Scholar

[23]

A. Neumaier, Completely regular codes, Discrete Math., 106/107 (1992), 353-360. doi: 10.1016/0012-365X(92)90565-W.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Rifà and V. A. Zinoviev, On lifting perfect codes,, preprint, ().   Google Scholar

[27]

N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Class of maximal equidistant codes, Problems Inform. Transmiss., 5 (1969), 84-87.  Google Scholar

[28]

N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Uniformly close-packed codes, Problems Inform. Transmiss., 7 (1971), 38-50.  Google Scholar

[29]

J. Singer, A theorem in finite projective geometry, and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.  Google Scholar

[30]

P. Solé, Completely regular codes and completely transitive codes, Discrete Math., 81 (1990), 193-201. doi: 10.1016/0012-365X(90)90152-8.  Google Scholar

[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.  Google Scholar

[32]

H. C. A. Van Tilborg, "Uniformly Packed Codes," Ph.D thesis, Eindhoven Univ. of Tech., 1976.  Google Scholar

[33]

V. A. Zinoviev and V. K. Leontiev, The nonexistence of perfect codes over Galois fields, Problems Control Inform. Th., 2 (1973), 16-24. Google Scholar

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.  Google Scholar

[2]

L. A. Bassalygo and V. A. Zinoviev, A remark on uniformly packed codes, Problems Inform. Transmiss., 13 (1977), 22-25.  Google Scholar

[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.  Google Scholar

[4]

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Borges, J. Rifà and V. A. Zinoviev, On linear completely regular codes with covering radius $\rho=1$,, preprint, ().   Google Scholar

[9]

A. E. Brouwer, A. M. Cohen and A. Neumaier, "Distance-Regular Graphs," Springer-Verlag, Berlin, 1989.  Google Scholar

[10]

K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426-434. doi: 10.1214/aoms/1177729387.  Google Scholar

[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.  Google Scholar

[12]

C. J. Colbourn and J. H. Dinitz, "The CRC Handbook of Combinatorial Designs," CRC Press, Boca Raton, FL, 1996. doi: 10.1201/9781420049954.  Google Scholar

[13]

G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, "Covering Codes," Elsevier Science, The Nederlands, 1997.  Google Scholar

[14]

P. Delsarte, Two-weight linear codes and strongly regular graphs, MBLE Research Laboratory, Report R160, 1971. Google Scholar

[15]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, 10 (1973), vi+97.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. M. Goethals and H. C. A. Van Tilborg, Uniformly packed codes, Philips Res., 30 (1975), 9-36.  Google Scholar

[20]

J. H. Koolen, W. S. Lee and W. J. Martin, Arithmetic completely regular codes,, preprint, ().   Google Scholar

[21]

F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Techn. J., 42 (1963), 79-84.  Google Scholar

[22]

F. J. MacWilliams and N. J. A. Sloane, "The Theory if Error-Correcting Codes," Elsevier, North-Holland, 1977.  Google Scholar

[23]

A. Neumaier, Completely regular codes, Discrete Math., 106/107 (1992), 353-360. doi: 10.1016/0012-365X(92)90565-W.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Rifà and V. A. Zinoviev, On lifting perfect codes,, preprint, ().   Google Scholar

[27]

N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Class of maximal equidistant codes, Problems Inform. Transmiss., 5 (1969), 84-87.  Google Scholar

[28]

N. V. Semakov, V. A. Zinoviev and G. V. Zaitsev, Uniformly close-packed codes, Problems Inform. Transmiss., 7 (1971), 38-50.  Google Scholar

[29]

J. Singer, A theorem in finite projective geometry, and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.  Google Scholar

[30]

P. Solé, Completely regular codes and completely transitive codes, Discrete Math., 81 (1990), 193-201. doi: 10.1016/0012-365X(90)90152-8.  Google Scholar

[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.  Google Scholar

[32]

H. C. A. Van Tilborg, "Uniformly Packed Codes," Ph.D thesis, Eindhoven Univ. of Tech., 1976.  Google Scholar

[33]

V. A. Zinoviev and V. K. Leontiev, The nonexistence of perfect codes over Galois fields, Problems Control Inform. Th., 2 (1973), 16-24. Google Scholar

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