General Galois Geometries

Hirschfeld, James Wp; Thas, J. A.

doi:10.1007/978-1-4471-6790-7

Cited by 307 publications

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“…Remark 1. After the paper was written, we found that Lemma 3 can also be obtained as a consequence of [21,Theorem 24.2.9,p.113] which discuss the theory of finite projective geometries. But, the proof of this Theorem requires more detailed theory which precedes it, while our proof is much shorter, simpler, and direct.…”

Section: Theorem 15 For Every Integer N ≥ 3 There Exists An 1-intermentioning

confidence: 99%

Equidistant codes in the Grassmannian

Etzion

Raviv

2015

Discrete Applied Mathematics

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Equidistant codes over vector spaces are considered. For k-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Plücker embedding, for 1-intersecting codes of k-dimensional subspaces over F n q , n ≥ k+1 2 , where the code size isis presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size n × n 2 over F q , rank n − 1, and rank distance n − 1.

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Section: Theorem 15 For Every Integer N ≥ 3 There Exists An 1-intermentioning

confidence: 99%

Equidistant codes in the Grassmannian

Etzion

Raviv

2015

Discrete Applied Mathematics

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“…It appears difficult to locate a result in Corollary 4.9 in standard treatises on finite geometry such as [11]. It can, however, be deduced from a relatively recent result of Kantor …”

Section: Projective Reed-muller Codesmentioning

confidence: 99%

Arithmetic, Geometry, Cryptography and Coding Theory

Kohel

Rolland

2017

Contemporary Mathematics

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Abstract. Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective space. We give a self-contained proof to show that this conjecture holds in the affirmative in the case of systems of three homogeneous polynomials, and also to show that the conjecture is false in the case of five quadrics in the 3-dimensional projective space over a finite field. Connections between the Tsfasman-Boguslavsky Conjecture and the determination of generalized Hamming weights of projective Reed-Muller codes are outlined and these are also exploited to show that this conjecture holds in the affirmative in the case of systems of a "large" number of three homogeneous polynomials, and to deduce the counterexample of 5 quadrics. An application to the nonexistence of lines in certain Veronese varieties over finite fields is also included.

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“…. ., rank-d subspaces of V (d + 1, q); for q = 2, this projective space features 2 d+1 − 1 points and (2 d+1 − 1)(2 d − 1)/3 lines (see, e. g., [14]). Given a PG(2N − 1, q) that is endowed with a non-degenerate symplectic form, the symplectic polar space W (2N −1, q) in PG(2N −1, q) is the space of all totally isotropic subspaces with respect to the non-degenerate symplectic form [15], with its maximal totally isotropic subspaces, also called generators, having dimension N − 1.…”

Section: Basic Concepts and Notationmentioning

confidence: 99%

Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

Санига

Holweck

Pracna

2017

Int. J. Geom. Methods Mod. Phys.

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Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type D n , 4 ≤ n ≤ 8, it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG(3, 2). Proper labelling of the vertices of the diagram (for 4 ≤ n ≤ 7) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG(3, 2). The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space W (3, 2) of the PG(3, 2) whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin-Peres magic square. In the case of n = 8, a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG(5, 2), the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG(3, 2) are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin-Peres square. Moreover, save for n = 4, each Veldkamp space is also endowed with some 'exceptional' point(s). Interestingly, two such points in the n = 8 case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix Y .

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General Galois Geometries

Cited by 307 publications

References 0 publications

Equidistant codes in the Grassmannian

Equidistant codes in the Grassmannian

Arithmetic, Geometry, Cryptography and Coding Theory

Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

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