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On determining all codes in semi-simple group rings. (English) Zbl 0804.94017
Cohen, Gérard (ed.) et al., Applied algebra, algebraic algorithms and error-correcting codes. 10th International symposium, AAECC-10, San Juan de Puerto Rico, Puerto Rico, May 10-14, 1993. Proceedings. Berlin: Springer-Verlag. Lect. Notes Comput. Sci. 673, 279-290 (1993).
Summary: A group algebra code is an ideal in a group ring, $$FG$$. If $$\text{char} (F)\nmid \text{order}(G)$$, then $$FG$$ is semi-simple and every such code has an idempotent generator. The group ring and every group algebra code in it are direct sums of disjoint minimal ideals. Thus a list of all possible codes in $$FG$$ may be produced by first determining idempotent generators of minimal codes which are direct summands of $$FG$$. When $$G$$ is abelian, the task is straightforward and is facilitated by use of the character table for $$G$$. When $$G$$ is non-abelian, however, two-sided ideals, which correspond to the group characters of $$G$$, in $$FG$$ may decompose in multiple ways to one-sided ideals. The decomposition may be varied by altering the matrix representations afforded by the group representations. The number of ways the representation may be altered is limited by the structure of the representation space. With sufficient information about this space, all possible decompositions can be determined. A case study is presented in which a $$(125,20)$$ two-sided ideal is found to contain 162 disjoint minimal left ideals, each with a distinct weight distribution.
For the entire collection see [Zbl 0823.00023].

MSC:
 94B15 Cyclic codes 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16D25 Ideals in associative algebras 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras