Extension sets, affine designs, and Hamada’s conjecture

Extension sets, affine designs, and Hamada’s conjecture We introduce the notion of an extension set for an affine plane of order q to study affine designs $${\mathcal {D}}'$$ D ′ with the same parameters as, but not isomorphic to, the classical affine design $${\mathcal {D}} = \mathrm {AG}_2(3,q)$$ D = AG 2 ( 3 , q ) formed by the points and planes of the affine space $$\mathrm {AG}(3,q)$$ AG ( 3 , q ) which are very close to this geometric example in the following sense: there are blocks $$B'$$ B ′ and B of $${\mathcal {D}'}$$ D ′ and $${\mathcal {D}}$$ D , respectively, such that the residual structures $${\mathcal {D}}'_{B'}$$ D B ′ ′ and $${\mathcal {D}}_B$$ D B induced on the points not in $$B'$$ B ′ and B, respectively, agree. Moreover, the structure $${\mathcal {D}}'(B')$$ D ′ ( B ′ ) induced on $$B'$$ B ′ is the q-fold multiple of an affine plane $${\mathcal {A}}'$$ A ′ which is determined by an extension set for the affine plane $$B \cong AG(2,q)$$ B ≅ A G ( 2 , q ) . In particular, this new approach will result in a purely theoretical construction of the two known counterexamples to Hamada’s conjecture for the case $$\mathrm {AG}_2(3,4)$$ AG 2 ( 3 , 4 ) , which were discovered by Harada et al. [7] as the result of a computer search; a recent alternative construction, again via a computer search, is in [23]. On the other hand, we also prove that extension sets cannot possibly give any further counterexamples to Hamada’s conjecture for the case of affine designs with the parameters of some $$\mathrm {AG}_2(3,q)$$ AG 2 ( 3 , q ) ; thus the two counterexamples for $$q=4$$ q = 4 might be truly sporadic. This seems to be the first result which establishes the validity of Hamada’s conjecture for some infinite class of affine designs of a special type. Nevertheless, affine designs which are that close to the classical geometric examples are of interest in themselves, and we provide both theoretical and computational results for some particular types of extension sets. Specifically, we obtain a theoretical construction for one of the two affine designs with the parameters of $$\mathrm {AG}_2(3,3)$$ AG 2 ( 3 , 3 ) and 3-rank 11 and for an affine design with the parameters of $$\mathrm {AG}_2(3,4)$$ AG 2 ( 3 , 4 ) and 2-rank 17 (in both cases, just one more than the rank of the classical example). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

Extension sets, affine designs, and Hamada’s conjecture

Loading next page...
 
/lp/springer_journal/extension-sets-affine-designs-and-hamada-s-conjecture-WnFCy0SoWn
Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media New York
Subject
Mathematics; Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits
ISSN
0925-1022
eISSN
1573-7586
D.O.I.
10.1007/s10623-017-0344-6
Publisher site
See Article on Publisher Site

Abstract

We introduce the notion of an extension set for an affine plane of order q to study affine designs $${\mathcal {D}}'$$ D ′ with the same parameters as, but not isomorphic to, the classical affine design $${\mathcal {D}} = \mathrm {AG}_2(3,q)$$ D = AG 2 ( 3 , q ) formed by the points and planes of the affine space $$\mathrm {AG}(3,q)$$ AG ( 3 , q ) which are very close to this geometric example in the following sense: there are blocks $$B'$$ B ′ and B of $${\mathcal {D}'}$$ D ′ and $${\mathcal {D}}$$ D , respectively, such that the residual structures $${\mathcal {D}}'_{B'}$$ D B ′ ′ and $${\mathcal {D}}_B$$ D B induced on the points not in $$B'$$ B ′ and B, respectively, agree. Moreover, the structure $${\mathcal {D}}'(B')$$ D ′ ( B ′ ) induced on $$B'$$ B ′ is the q-fold multiple of an affine plane $${\mathcal {A}}'$$ A ′ which is determined by an extension set for the affine plane $$B \cong AG(2,q)$$ B ≅ A G ( 2 , q ) . In particular, this new approach will result in a purely theoretical construction of the two known counterexamples to Hamada’s conjecture for the case $$\mathrm {AG}_2(3,4)$$ AG 2 ( 3 , 4 ) , which were discovered by Harada et al. [7] as the result of a computer search; a recent alternative construction, again via a computer search, is in [23]. On the other hand, we also prove that extension sets cannot possibly give any further counterexamples to Hamada’s conjecture for the case of affine designs with the parameters of some $$\mathrm {AG}_2(3,q)$$ AG 2 ( 3 , q ) ; thus the two counterexamples for $$q=4$$ q = 4 might be truly sporadic. This seems to be the first result which establishes the validity of Hamada’s conjecture for some infinite class of affine designs of a special type. Nevertheless, affine designs which are that close to the classical geometric examples are of interest in themselves, and we provide both theoretical and computational results for some particular types of extension sets. Specifically, we obtain a theoretical construction for one of the two affine designs with the parameters of $$\mathrm {AG}_2(3,3)$$ AG 2 ( 3 , 3 ) and 3-rank 11 and for an affine design with the parameters of $$\mathrm {AG}_2(3,4)$$ AG 2 ( 3 , 4 ) and 2-rank 17 (in both cases, just one more than the rank of the classical example).

Journal

Designs, Codes and CryptographySpringer Journals

Published: Mar 1, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off