Study of idempotents in cyclic group rings over F2

Ong, Kai Lin; Ang, Miin Huey

doi:10.1063/1.4952491

Cited by 5 publications

(6 citation statements)

References 5 publications

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“…Note that result below further assure that the generated idempotents are generalization from our basis idempotents in [7]. (2,4), (1,5), (2,7), (1, 8)}. By Theorem 3.5, we yield the corresponding generated idempotents:…”

Section: Identification Of All Idempotentsmentioning

confidence: 66%

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Full Identification of Idempotens in Binary Abelian Group Rings

Ong¹,

Ang²

2017

JIMS

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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.

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Section: Identification Of All Idempotentsmentioning

confidence: 66%

“…In [7], we have fully identify all idempotents in every F 2 C n algebraically, where C n is a cyclic group, by using our concept of basis idempotents. We show that every F 2 C n will have a unique idempotent, e L with largest support size.…”

Section: Introductionmentioning

confidence: 99%

Full Identification of Idempotens in Binary Abelian Group Rings

Ong¹,

Ang²

2017

JIMS

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“…This section is mainly devoted to studying the algebraic structure of the set of all idempotents in G F(4)G for finite abelian group G, denoted by I G F(4) (G). Since char(G F( 4)) = 2 = char(F 2 ), the approach used in developing the result in this session will be similar to those in [23,24].…”

Section: Idempotents In Commutative Group Algebra Over Gf(4)mentioning

confidence: 99%

Burst error-correcting quantum stabilizer codes designed from idempotents

Ong

2023

Quantum Inf Process

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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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“…This lead to the conjecture "Every group ring code in F 2 D 2n is equivalent to some group ring codes in F 2 C 2n " [13]. As any arbitrary idempotent in F 2 G can be chosen as a generator of a zero-divisor code C = W u in F 2 G, the primary step to study equivalency is to establish a way to classify and identify idempotents in F 2 G. Let I (G) be the set of all idempotents in F 2 G. For abelian G, a new classification of idempotents that completely classified and identified all elements in I (G) were introduced by Ong and Ang in 2017 [9,10]. This was done by first affirming the existence of a proper subset of I (G) where the support of each of its element is generated by an element in G. for g ∈ supp(e), then e is said to be a generated idempotent in F 2 G with generator g and is denoted by g I d .…”

Section: Definition 12mentioning

confidence: 99%

“…Readers who are interested with the details of the complete identification of I I d (G) for abelian G are referred to [9,10]. We include a concise example to illustrate the identification of I (G) for an abelian G. Note that, throughout this paper, C n denotes a cyclic group of order n whereas C g denotes a cyclic group with generator g. Example 1.…”

Section: Theorem 14 Let G Be An Abelian Group Then I I D (G) Is a Basis For I (G)mentioning

confidence: 99%

On equivalency of zero-divisor codes via classifying their idempotent generator

Ong¹,

Ang²

2020

Des. Codes Cryptogr.

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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.

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Study of idempotents in cyclic group rings over F2

Cited by 5 publications

References 5 publications

Full Identification of Idempotens in Binary Abelian Group Rings

Full Identification of Idempotens in Binary Abelian Group Rings

Burst error-correcting quantum stabilizer codes designed from idempotents

On equivalency of zero-divisor codes via classifying their idempotent generator

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