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

2010; American Institute of Mathematical Sciences; Volume: 4; Issue: 4 Linguagem: Inglês

10.3934/amc.2010.4.567

1930-5346

Joaquim Borges, Josep Rifà, Victor Zinoviev,

Finite Group Theory Research

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.