May  2018, 12(2): 337-349. doi: 10.3934/amc.2018021

Completely regular codes by concatenating Hamming codes

1. 

Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, Spain

2. 

A.A. Kharkevich Institute for Problems of Information Transmission, Russian Academy of Sciences, Russia

Received  March 2017 Revised  July 2017 Published  March 2018

Fund Project: This work has been partially supported by the Spanish grants TIN2016-77918-P, AEI/FEDER, UE., MTM2015-69138-REDT; and also by Russian Foundation for Sciences (14-50-00150).

We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of these codes. As a result, we find some non-equivalent completely regular codes, over the same finite field, with the same parameters and intersection array. We also study when the extension of these codes gives completely regular codes. Some of these new codes are completely transitive.

Citation: Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021
References:
[1]

E. AssmusJ. M. Goethals and H. Mattson, Generalized t-designs and majority decoding of linear codes, Information and Control, 32 (1976), 43-60. doi: 10.1016/S0019-9958(76)90101-7. Google Scholar

[2]

L. A. BassalygoG. V. Zaitsev and V. A. Zinoviev, Zinoviev, Uniformly packed codes, Problems Inform. Transmiss, 10 (1974), 9-14. Google Scholar

[3]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, vol. 69, Cambridge University Press, Cambridge, 1986. Google Scholar

[4]

I. F. Blake and R. C. Mullin, The Mathematical Theory of Coding, New York-London, 1975. Google Scholar

[5]

A. E. Brouwer, On complete regularity of extended codes, Discrete Mathematics, 117 (1993), 271-273. doi: 10.1016/0012-365X(93)90342-Q. Google Scholar

[6]

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

[7]

R. Calderbank and W. Kantor, The geometry of two-weight codes, Bulletin of the London Mathematical Society, 18 (1986), 97-122. doi: 10.1112/blms/18.2.97. Google Scholar

[8]

P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Thesis, 1973. Google Scholar

[9]

M. Giudici and C. E. Praeger, Completely transitive codes in hamming graphs, European Journal of Combinatorics, 20 (1999), 647-661. doi: 10.1006/eujc.1999.0313. Google Scholar

[10]

J. Goethals and H. VanTilborg, Uniformly packed codes, Philips Research Reports, 30 (1975), 9-36. Google Scholar

[11]

D. Hughes and F. Piper, Design Theory, Cambridge University Press, 1985. Google Scholar

[12]

J. Koolen, D. Krotov and B. Martin, Completely regular codes, https://sites.google.com/site/completelyregularcodes.Google Scholar

[13]

K. Lindström, All nearly perfect codes are known, Information and Control, 35 (1977), 40-47. doi: 10.1016/S0019-9958(77)90519-8. Google Scholar

[14]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, 1977. Google Scholar

[15]

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

[16]

J. Rifà and V. Zinoviev, Completely regular codes with different parameters giving the same distance-regular coset graphs, Discrete mathematics, 340 (2017), 1649-1656. doi: 10.1016/j.disc.2017.03.001. Google Scholar

[17]

N. SemakovV. A. Zinoviev and G. Zaitsev, Uniformly packed codes, Problemy Peredachi Informatsii, 7 (1971), 38-50. Google Scholar

[18]

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

[19]

E. R. van Dam, J. H. Koolen and H. Tanaka, Distance-regular graphs, The Electronic Journal of Combinatorics, #DS22, 1st edition (2016), 1–156.Google Scholar

[20]

H. C. A. van Tilborg, Uniformly Packed Codes, Technische Hogeschool Eindhoven, 1976. Google Scholar

[21]

V. Zinoviev and J. Rifá, On new completely regular q-ary codes, Problems of Information Transmission, 43 (2007), 97-112. Google Scholar

show all references

References:
[1]

E. AssmusJ. M. Goethals and H. Mattson, Generalized t-designs and majority decoding of linear codes, Information and Control, 32 (1976), 43-60. doi: 10.1016/S0019-9958(76)90101-7. Google Scholar

[2]

L. A. BassalygoG. V. Zaitsev and V. A. Zinoviev, Zinoviev, Uniformly packed codes, Problems Inform. Transmiss, 10 (1974), 9-14. Google Scholar

[3]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, vol. 69, Cambridge University Press, Cambridge, 1986. Google Scholar

[4]

I. F. Blake and R. C. Mullin, The Mathematical Theory of Coding, New York-London, 1975. Google Scholar

[5]

A. E. Brouwer, On complete regularity of extended codes, Discrete Mathematics, 117 (1993), 271-273. doi: 10.1016/0012-365X(93)90342-Q. Google Scholar

[6]

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

[7]

R. Calderbank and W. Kantor, The geometry of two-weight codes, Bulletin of the London Mathematical Society, 18 (1986), 97-122. doi: 10.1112/blms/18.2.97. Google Scholar

[8]

P. Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Thesis, 1973. Google Scholar

[9]

M. Giudici and C. E. Praeger, Completely transitive codes in hamming graphs, European Journal of Combinatorics, 20 (1999), 647-661. doi: 10.1006/eujc.1999.0313. Google Scholar

[10]

J. Goethals and H. VanTilborg, Uniformly packed codes, Philips Research Reports, 30 (1975), 9-36. Google Scholar

[11]

D. Hughes and F. Piper, Design Theory, Cambridge University Press, 1985. Google Scholar

[12]

J. Koolen, D. Krotov and B. Martin, Completely regular codes, https://sites.google.com/site/completelyregularcodes.Google Scholar

[13]

K. Lindström, All nearly perfect codes are known, Information and Control, 35 (1977), 40-47. doi: 10.1016/S0019-9958(77)90519-8. Google Scholar

[14]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, 1977. Google Scholar

[15]

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

[16]

J. Rifà and V. Zinoviev, Completely regular codes with different parameters giving the same distance-regular coset graphs, Discrete mathematics, 340 (2017), 1649-1656. doi: 10.1016/j.disc.2017.03.001. Google Scholar

[17]

N. SemakovV. A. Zinoviev and G. Zaitsev, Uniformly packed codes, Problemy Peredachi Informatsii, 7 (1971), 38-50. Google Scholar

[18]

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

[19]

E. R. van Dam, J. H. Koolen and H. Tanaka, Distance-regular graphs, The Electronic Journal of Combinatorics, #DS22, 1st edition (2016), 1–156.Google Scholar

[20]

H. C. A. van Tilborg, Uniformly Packed Codes, Technische Hogeschool Eindhoven, 1976. Google Scholar

[21]

V. Zinoviev and J. Rifá, On new completely regular q-ary codes, Problems of Information Transmission, 43 (2007), 97-112. Google Scholar

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