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Definition
A point-weight incidence structure is an incidence structure where every point is associated with an positive integer weight. Formally, a point-weight design is a quadruple (V,B,I,w) where (V,B,I) is an incidence structure and w is a map from the set of points V into the positive integers.
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Here
V={P1,P2,P3,P4}
B={B1,B2,B3,B4}
and
w(P1)=1
w(P2)=1
w(P3)=1
w(P4)=2
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A point-weight design is an extension of the idea of a design into a point-weight incidence structure. A point-weight design is characterised by two conditions: the constant block size condition and the design condition. In a classical design the constant block size condition is "every block must contain exactly k points" and this easily generalises to point-weight incidence structures. We will require that "the sum of the weights of the points on a block is constant". Notice that the above point-weight incidence structure has the constant block size property as the sum of the weights on any block is 3.
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The constant block size condition is quite obvious but there is more than one consistant definition of the design condition. (By "consistant" we mean that if some structure is a point-weight design where all the points are the same weight then it is essentially a classical design). The most obvious is a design condition that has any t points lie on exactly  blocks. A point-weight incidence structure that satisfies both this design condition and the constant block size condition is called a point-sum point-weight design and is dealt with extensively in the PhD thesis of Richard Horne (here). The above is a example is a point-sum design for t=2 and =1.
Another design condition, proposed by Tracey Powlesland in her thesis, is that every set of points whose weights add up to t should lie on exactly blocks. A point-weight incidence structure that satisfies both this design condition and the constant block size condition is called a weight-sum point-weight design. The above example is, rather trivially, a weight-sum design for t=3 and =1. Weight-sum designs have applications to secret sharing schemes and are therefore the most difficult type of point-weight design for which to derive abstract results!
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The last studied design condition was studied in my thesis (here). If S is a set of t points then these points are incident with exactly
 .
This somewhat bizarre design condition gives rise to certain desirable properties of the incidence matrix of the design. We a point-weight incidence structure that satisfies both this design condition and the constant block size condition is called a row-sum point-weight design. The design on the right is an example of a row-sum design for t=2 and =4.
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