Browsing by Subject "Group codes"

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    An intrinsical description of group codes
    (Springer, 2009-01-06) Bernal Buitrago, José Joaquín; Río Mateos, Ángel del; Simón Pinero, Juan Jacobo; Matemáticas
    A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism FG → Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space Fn, which does not assume an “a priori” group algebra structure on Fn. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.
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    Group code structures of affine-invariant codes
    (2011-01) Bernal Buitrago, José Joaquín; Río Mateos, Ángel del; Simón Pinero, Juan Jacobo; Matemáticas
    A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a nite group. In these realizations, the group algebra is identi ed with the ambient space, and the group elements with the coordinates of the ambient space. It is an obvious consequence of the de nition that every pr-ary a ne-invariant code of length pm, with p prime, can be realized as an ideal of the group algebra Fpr [(Fpm; +)], where (Fpm; +) is the underlying additive group of the eld Fpm with pm elements. In this paper we describe all the group code structures of an a ne-invariant code of length pm in terms of a family of maps from Fpm to the group of automorphisms of (Fpm; +). We also present a familly of non-obvious group code structures in an arbitrary a ne-invariant code.