Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent.
In 2009, Ted and Paul Hurley proposed a code construction method using group rings. These codes with single generator are termed group ring codes and in particular zero-divisor codes when using zero-divisors as generators. In this paper, we mainly study the equivalency of zero-divisor codes in F 2 G having generator from I (G), the set of all idempotents in F 2 G. For abelian G, our previous notion of generated idempotents completely classified I (G) by serving as its basis. Here, we first extend the notion of generated idempotents to study and classify some elements in I (G) for non-abelian G. Later, the study is generally done on equivalency of zero-divisor codes in F 2 G, then concentrating on those with idempotent generator. In particular, we affirm the conjecture "Every group ring code in F 2 D 2n is equivalent to some in F 2 C 2n " in the cases where the generators are our classified idempotents. We also show that the equivalency of zero-divisor codes in F 2 C n with generated idempotent as generators can be established sufficiently on the generator property.
Zero-divisor codes are codes constructed using group rings where their generators are zero-divisors. Generally, zero-divisor codes can be equivalent despite their associated groups are non-isomorphic, leading to the proposed conjecture “Every dihedral zero-divisor code has an equivalent form of cyclic zero-divisor code”. This paper is devoted to study equivalence of zero-divisor codes in $$F_2G$$ F 2 G having generators from the 2-nilradical of $$F_2G$$ F 2 G , consisting of all nilpotents of nilpotency degree 2 of $$F_2G$$ F 2 G . Essentially, algebraic structures of 2-nilradicals are first studied in general for both commutative and non-commutative $$F_2G$$ F 2 G before specialized into the case when G is cyclic and dihedral. Then, results are used to study the conjecture above in the cases where the codes generators are from their respective 2-nilradicals.
Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially $$F_2$$ F 2 -submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group $$C_n$$ C n , for $$n=2^m-1$$ n = 2 m - 1 and $$n=2^m+1$$ n = 2 m + 1 . Quantum bounds on their burst error minimum distance are subsequently determined.
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