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

Joaquim Borges; Josep Rifà; Victor A. Zinoviev

doi:10.3934/amc.2010.4.567

Article Contents

2010, Volume 4, Issue 4: 567-578. Doi: 10.3934/amc.2010.4.567

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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
2.
Institute for Problems of Information Transmission, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994

Received: January 31, 2010

Published: October 2010

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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.

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Mathematics Subject Classification: Primary: 94B25; Secondary: 94B60.

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