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On $q$-ary linear completely regular codes with $\rho=2$ and antipodal dual
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 |
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, ().
|
[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, ().
|
[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, ().
|
[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, ().
|
[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, ().
|
[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, ().
|
[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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