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

Pages: 567 - 578, Volume 4, Issue 4, November 2010      doi:10.3934/amc.2010.4.567

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Joaquim Borges - Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain (email)
Josep Rifà - Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain (email)
Victor A. Zinoviev - Institute for Problems of Information Transmission, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russian Federation (email)

Abstract: 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:  Linear completely regular codes, completely transitive codes, covering radius.
Mathematics Subject Classification:  Primary: 94B25; Secondary: 94B60.

Received: February 2010;      Available Online: November 2010.

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