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Metacyclic error-correcting codes. (English) Zbl 0823.94017
The group codes are ideals of group algebras. Let \(G\) and \(H\) be groups of the same order, \(F\) be a finite field, let \(FG\) and \(FH\) be the corresponding group rings. The combinatorial equivalence is an \(F\) vector space isomorphism \(\gamma: FG\to FH\) induced by a bijection \(\gamma: G\to H\). Codes \(C1 \subseteq FG\) and \(C2 \subseteq FH\) are said to be combinatorially equivalent if there exists a combinatorial equivalence \(\gamma: FG\to FH\) such that \(\gamma (C1)= C2\).
In this paper the authors investigate the group codes when the underlying groups are metacyclic groups of odd order and \(F\) is a field of characteristic two. Using methods of representation theory it is shown that each of these codes is combinatorially equivalent to an abelian code, and therefore they are not interesting as non-abelian codes. Certain of them contain one-sided (which are one-sided ideals) non- abelian codes which are not combinatorially equivalent and may vary in minimal distance. There are also given examples, a list of the better metacyclic codes and the algorithm to determine minimal left metacyclic codes.

94B15 Cyclic codes
20G15 Linear algebraic groups over arbitrary fields
94B65 Bounds on codes
Full Text: DOI
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