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).
Designs, Codes and Cryptography – Springer Journals
Published: Mar 1, 2017
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
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.
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.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”
Daniel C.
“Whoa! It’s like Spotify but for academic articles.”
@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”
@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”
@JoseServera
DeepDyve Freelancer | DeepDyve Pro | |
---|---|---|
Price | FREE | $49/month |
Save searches from | ||
Create lists to | ||
Export lists, citations | ||
Read DeepDyve articles | Abstract access only | Unlimited access to over |
20 pages / month | ||
PDF Discount | 20% off | |
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.
ok to continue