Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of François Jaeger (a survey)

Bannaĭ, Eiichi

doi:10.5802/aif.1690

Cited by 8 publications

(7 citation statements)

References 19 publications

(31 reference statements)

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“…The octahedral group is not of the reflection type but is isomorphic to U 4 [22]. It possesses the invariant polynomial (9) for the vertices and another invariant polynomial related to the 8 faces of the octahedron centered at the points given in (10).…”

Section: The Octahedral Groupmentioning

confidence: 99%

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On the Geometry and Invariants of Qubits, Quartits and Octits

Planat

2011

Int. J. Geom. Methods Mod. Phys.

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Four level quantum systems, known as quartits, and their relation to twoqubit systems are investigated group theoretically. Following the spirit of Klein's lectures on the icosahedron and their relation to Hopf sphere fibrations, invariants of complex reflection groups occuring in the theory of qubits and quartits are displayed. Then, real gates over octits leading to the Weyl group of E 8 and its invariants are derived. Even multilevel systems are of interest in the context of solid state nuclear magnetic resonance.

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Section: The Octahedral Groupmentioning

confidence: 99%

“…Primary invariants T and W, the secondary invariant κ and relation(22) follow from Magma with the following code "R:=InvariantRing(U8); PrimaryInvariants(R); Sec-ondaryInvariants(R); Algebra(R); Relations(R);"…”

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confidence: 99%

On the Geometry and Invariants of Qubits, Quartits and Octits

Planat

2011

Int. J. Geom. Methods Mod. Phys.

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“…Since the coefficients of any weight enumerators of codes are in Z, a natural question is if any weight enumerator of all doubly even self-dual binary codes can be written as a polynomial in W (1) e 8 and W (1) g 24 over a smaller ring than C. It turns out that we can replace C in the equality in equation (1.1) by the smaller ring Z…”

Section: Introductionmentioning

confidence: 99%

“…For instance, it is well known that the weight enumerator of every doubly even self-dual binary code is a polynomial in two generators, the complete weight enumerator W (1) e 8 of the Hamming code and the complete weight enumerator W (1) g 24 of the Golay code. The graded ring C[W (1) C ] of the weight enumerators of all doubly even self dual binary codes is isomorphic to the graded ring C[E 4 , ∆] of elliptic modular forms [3]; explicitly,…”

Section: Introductionmentioning

confidence: 99%

The joint weight enumerators and Siegel modular forms

Choie

Oura

2006

Proc. Amer. Math. Soc.

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“…Sometimes, we can even consider the codes over certain finite rings and finite Abelian groups. A brief survey on this subject can be seen, for example, in [1]. In [1], we define Type II codes (the concept corresponding to self-dual doubly even codes in binary case) over any finite Abelian group of even order, by using the concept of duality and the modular invariance property for (the association schemes of) finite Abelian groups, which can be seen in Bannai et al [3].…”

Section: Introductionmentioning

confidence: 99%

On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

Bannaĭ

Ozeki

Teranishi

1999

European Journal of Combinatorics

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We construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason-MacWilliams group. We find this canonical set in the vector space (⊗ l i=1 V ) G , where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C[x, y] G 0 of the two-dimensional group G 0 of order 48 which is the intersection of G and S L(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group S l . That is, by letting the symmetric group S l acts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group S l acts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗ l i=1 V ) G . This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms.

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Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of François Jaeger (a survey)

Cited by 8 publications

References 19 publications

On the Geometry and Invariants of Qubits, Quartits and Octits

On the Geometry and Invariants of Qubits, Quartits and Octits

The joint weight enumerators and Siegel modular forms

On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

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