More Results on the Existence of Simple BIBDs with Number of Elements a Prime Power

We study the existence of simple (q, k, λ) BIBDs when the number of elements is a prime powerqand {c 1,c 2} ⋂ {1, 2} is not empty, wherec 1= gcd(k, q− 1) andc 2= gcd(k− 1,q− 1). We show that in many situations the necessary conditions λ(q− 1) ≡ 0 mod (k− 1), λq(q− 1) ≡ 0 modk(k− 1), and λ ≤ ( k − 2...

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Bibliographic Details
Published inTaiwanese journal of mathematics Vol. 20; no. 3; pp. 523 - 543
Main Author Sun, Hsin-Min
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.06.2016
Subjects
Equivalence relation
Existence theorems
Integers
Necessary conditions
Prime numbers
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Summary:We study the existence of simple (q, k, λ) BIBDs when the number of elements is a prime powerqand {c 1,c 2} ⋂ {1, 2} is not empty, wherec 1= gcd(k, q− 1) andc 2= gcd(k− 1,q− 1). We show that in many situations the necessary conditions λ(q− 1) ≡ 0 mod (k− 1), λq(q− 1) ≡ 0 modk(k− 1), and λ ≤ ( k − 2 q − 2 ) are also sufficient for the existence of a simple (q, k, λ) BIBD. These new results improve the valid range of simple BIBDs. 2010Mathematics Subject Classification. Primary 05B05; Secondary 12E20. Key words and phrases. Balanced incomplete block design, Finite field, Generating block.
ISSN:1027-5487
2224-6851
DOI:10.11650/tjm.20.2016.5021