Results 1 to 2 of 2

Thread: Steiner-Triple Systems

  1. #1
    Oct 2010

    Steiner-Triple Systems

    There was no combinatorics section so this is probably the best section to post this question in...

    I was looking at how to construct STS(n) where p=1 mod 6 and p prime.
    The following method I know works but I was just wondering why it works... if someone could explain that would be great:

    Find w such that w^3=1 mod p
    Construct the set {1,w,w^2} which are the roots of positive unity...
    We can form a group {1,w,w^2,-1,-w,-(w^2)} with multiplication under mod p.
    This is the 6 roots of unity.

    Once we have done that we can find all the right cosets until we have exhausted the integers mod p. If p=6m+1 then we will have m cosets... Now we take the first three elements of each set equivalent to 1,w and w^2 and make a new set. We will have m sets each with 3 elements.

    Property) Now, each non-zero element x of the integers mod p will have a unique solution u-v=x where u and v are in the same set that we have constructed. So if out sets of 3 are B1,B2,....BM we can construct the STS(p) be taking the set {B1+k,B2+k,...,BM+k:k is in integers mod p} which I get because of the property I explained. But how do we know constructing sets in this way will give this property?

    I hope I have explained it well enough.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Oct 2010
    Does anyone know?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Steiner Triple System
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Jun 7th 2011, 01:58 PM
  2. Replies: 0
    Last Post: Feb 17th 2011, 06:23 PM
  3. Replies: 0
    Last Post: Feb 13th 2011, 11:40 AM
  4. Triple Integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 11th 2009, 11:12 AM
  5. Jakob Steiner - Synthetic Geometry
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: Feb 9th 2008, 11:33 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags

/mathhelpforum @mathhelpforum