Projective two-weight codes with small parameters and their corresponding graphs

Bouyukliev, Iliya; Fack, Veerle; Willems, Wolfgang M.; Winne, Joost

doi:10.1007/s10623-006-0019-1

Cited by 35 publications

(63 citation statements)

References 13 publications

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“…The following problem was posed in 2001 by H. Ward [28], who had raised the special case q = p in 1998 [27]: determine whether or not there exists a two-weight 3q 2 +3 2 , 4, 3q 2 −3q 2 code C over GF (q) with second weight 3q 2 −q 2 . ; the problem is quoted in [2] in the form given on that list.) At the time the paper [28] was written, the existence of such a code was known for q = 3 [26] and for q = 5 [1,25], but not for q > 5.…”

Section: Theorem 1 (Griesmer Bound) Suppose There Exists An [N K Dmentioning

confidence: 99%

“…There are exactly two inequivalent such codes: a first [26] for which H is isomorphic to S 6 [6], and a second [14] for which H has order 36 [2].…”

Section: Theorem 1 (Griesmer Bound) Suppose There Exists An [N K Dmentioning

confidence: 99%

“…It can be shown that the set of points comprising the duals of the small planes of the projective set O of (4) forms another projective set with the same parameters as O; in the terminology of [2], it follows that the code C is formally projective self-dual (see also [27, p. 92]). We state and prove the following consequence of this result.…”

Section: Definitions and Notationmentioning

confidence: 99%

“…If P is a projective set then, by Theorem 2, we can interpret the points of P as the columns of a generator matrix for a projective two-weight code C. The automorphism group of C is isomorphic to the semidirect product of Aut(P ) with Gal(GF(q)/GF(p)) (where the Galois group takes account of the action of a field automorphism applied to all co-ordinate positions of the codewords; see [2] for further discussion). In the case q = p, which is our primary interest here, the automorphism group of C is isomorphic to Aut(P ).…”

Section: Definitions and Notationmentioning

confidence: 99%

“…The base plane is isomorphic to PG (2,7). By the definition of projective set, the base plane contains exactly 12 points of O, and by Proposition 4 no four of these points are collinear.…”

Section: Potential Base Plane Setsmentioning

confidence: 99%

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On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)

2010

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We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound.Mathematics Subject Classification (2010). 05E20, 05B25, 94B05.

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Section: Theorem 1 (Griesmer Bound) Suppose There Exists An [N K Dmentioning

confidence: 99%

“…There are exactly two inequivalent such codes: a first [26] for which H is isomorphic to S 6 [6], and a second [14] for which H has order 36 [2].…”

Section: Theorem 1 (Griesmer Bound) Suppose There Exists An [N K Dmentioning

confidence: 99%

Section: Definitions and Notationmentioning

confidence: 99%

Section: Definitions and Notationmentioning

confidence: 99%

“…The base plane is isomorphic to PG (2,7). By the definition of projective set, the base plane contains exactly 12 points of O, and by Proposition 4 no four of these points are collinear.…”

Section: Potential Base Plane Setsmentioning

confidence: 99%

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On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)

2010

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Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

Feng

Chen

2019

J of Combinatorial Designs

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In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.

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Partial Spreads and Vector Space Partitions

Honold

Kiermaier

Kurz

2018

Network Coding and Subspace Designs

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ABSTRACT. Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake & Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.

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Projective two-weight codes with small parameters and their corresponding graphs

Cited by 35 publications

References 13 publications

On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)

On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

Partial Spreads and Vector Space Partitions

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