The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs

Cara, Philippe; Rottey, Sara; Voorde, Geertrui Van de

doi:10.1515/advgeom-2013-0040

Cited by 5 publications

(1 citation statement)

References 3 publications

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“…incidence system is called the linear representation of the set K and is denoted by T * n−1 (K) (Figure 1). It has been studied for many choices of K in [4], and the automorphism group has been studied in [2].…”

Section: Introductionmentioning

confidence: 99%

Linear representations of finite geometries and associated LDPC codes

Sin

Sorci

Xiang

2020

Journal of Combinatorial Theory, Series A

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The linear representation of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes. We consider the LDPC codes defined by taking the incidence matrix and its transpose as parity-check matrices, and in the former case prove a conjecture of Vandendriessche that the code is generated by words of minimum weight called plane words. In the latter case we compute the minimum weight in several cases and provide explicit constructions of minimum weight codewords.

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Section: Introductionmentioning

confidence: 99%

Linear representations of finite geometries and associated LDPC codes

Sin

Sorci

Xiang

2020

Journal of Combinatorial Theory, Series A

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Projective Paley sets

Cossidente

Marino

Pavese

2019

J of Combinatorial Designs

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A two‐character set in PG(r,q) is a set scriptX of points with the property that the intersection number with any hyperplane only takes two values. A projective Paley set of PG(2n−1,q)goodbreakinfix,0.33emq odd, is a subset scriptX of points such that every hyperplane of PG(2n−1,q) meets scriptX in either (qn+1)(qn−1−1)∕2(q−1) or (qn−1)(qn−1+1)∕2(q−1) points. A quasi‐quadric in PG(2n−1,q) is a two‐character set that has the same size and the same intersection numbers with respect to hyperplanes as a nondegenerate quadric. Here we construct projective Paley sets of PG(3,q) left invariant by a cyclic group of order q2goodbreakinfix+1 and of PG(5,q) admitting PSL(2,q2) as an automorphism group. Also infinite families of quasi‐quadrics of PG(5,q) are provided.

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Linear representations of subgeometries

Winter

Rottey

Voorde

2014

Des. Codes Cryptogr.

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The linear representation T∗n(K) of a point set K in a hyperplane of PG(n+1,q) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations T∗n(K) and T∗n(K′), under a few conditions on K and K′. First, we prove that an isomorphism between T∗n(K) and T∗n(K′) is induced by an isomorphism between the two linear representations T∗n(K) and T∗n(K′) of their closures K and K′. This allows us to focus on the automorphism group of a linear representation T∗n(S) of a subgeometry S≅PG(n,q) embedded in a hyperplane of the projective space PG(n+1,qt). To this end we introduce a geometry X(n,t,q) and determine its automorphism group. The geometry X(n,t,q) is a straightforward generalization of Hn+2q which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of X(n,t,q) as a coset geometry we extend this result and prove that X(n,t,q) and T∗n(S) are isomorphic. Finally, we compare the full automorphism group of T∗n(S) with the “natural” group of automorphisms that is induced by the collineation group of its ambient space

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The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs

Cited by 5 publications

References 3 publications

Linear representations of finite geometries and associated LDPC codes

Linear representations of finite geometries and associated LDPC codes

Projective Paley sets

Linear representations of subgeometries

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