A packing problem its application to Bose's families
β Scribed by Marco Buratti
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 906 KB
- Volume
- 4
- Category
- Article
- ISSN
- 1063-8539
No coin nor oath required. For personal study only.
β¦ Synopsis
We propose and study the following problem: given X c z,, construct a maximum packing of devx (the development of X ) , i.e., a maximum set of pairwise disjoint translates of X. Such a packing is optimul when its size reaches the upper bound 1 & 1. In particular, it is perfect when its size is exactly equal to 5, i.e. when it is a partition of z,. We apply the above problem for constructing Bose's families. A (q, k) Bose's family (BF) is a nonempty family 3 of subsets of the field GF(y) such that: (i) each member of F is a coset of the kth roots of unity for k odd (the union of a coset of the (k -1)th roots of unity and zero for k even); (ii) the development of F, i.e., the incidence structure (GF(q), I3: = (X + gI(X, g ) E F x GF(q)), is a semilinear space. A (4, k)-BF is optimal when its size reaches the upper bound [a]. In particular, it is perfect when its size is exactly equal to &; in this case the (y, k)-BF is a (q, k, 1) difference family and its development is a linear space. If the set of (q, k)-BF's is not empty, there is a bijection preserving maximality, optimality, and perfectness between this set with the set of packings of devX, where X is a suitable [$]-subset of z,, n = $ for k odd, n = -for k even.
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