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Translation-invariant propelinear codes

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2 Author(s)
Rifa, J. ; Dept. d''Inf., Univ. Autonoma de Barcelona, Spain ; Pujol, J.

A class of binary group codes is investigated. These codes are the propelinear codes, defined over the Hamming metric space Fm, F=(0, 1), with a group structure. Generally, they are neither Abelian nor translation-invariant codes but they have good algebraic and combinatorial properties. Linear codes and Z4-linear codes can be seen as a subclass of propelinear codes. It is shown here that the subclass of translation-invariant propelinear codes is of type Z2k1⊕Z4k2⊕Q 8(k3) where Q8 is the non-Abelian quaternion group of eight elements. Exactly, every translation-invariant propelinear code of length n can be seen as a subgroup of Z2k1⊕Z4k2⊕Q 8k3 with k1+2k2+4k3 =n. For k2=k3=0 we obtain linear binary codes and for k1=k3=0 we obtain Z4-linear codes. The class of additive propelinear codes-the Abelian subclass of the translation-invariant propelinear codes-is studied and a family of nonlinear binary perfect codes with a very simply construction and a very simply decoding algorithm is presented

Published in:

Information Theory, IEEE Transactions on  (Volume:43 ,  Issue: 2 )

Date of Publication:

Mar 1997

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