A difference multiset in an additive group \$G\$ is a multiset \$X=(x_1,\cdots,x_k)\$ such that the list \$\Delta X=(x_i-x_j\:\ 1\le i\le k,\ 1\le j\le k,\ i\ne j)\$ contains each element (including 0) of \$G\$ the same number of times. Similarly, a strong difference family (SDF) is a family of multisets \$X_1,\cdots,X_t\$ such that the union of the lists \$\Delta X_i\$ contains each element of \$G\$ the same number of times. It is shown how useful these are in the construction of regular group divisible designs and regular or 1-rotational balanced incomplete block designs. Examples using Paley difference multisets or Paley SDFs are given, and many new resolvable 1-rotational designs are constructed.

### Old and new designs via difference multisets and strong difference families

#### Abstract

A difference multiset in an additive group \$G\$ is a multiset \$X=(x_1,\cdots,x_k)\$ such that the list \$\Delta X=(x_i-x_j\:\ 1\le i\le k,\ 1\le j\le k,\ i\ne j)\$ contains each element (including 0) of \$G\$ the same number of times. Similarly, a strong difference family (SDF) is a family of multisets \$X_1,\cdots,X_t\$ such that the union of the lists \$\Delta X_i\$ contains each element of \$G\$ the same number of times. It is shown how useful these are in the construction of regular group divisible designs and regular or 1-rotational balanced incomplete block designs. Examples using Paley difference multisets or Paley SDFs are given, and many new resolvable 1-rotational designs are constructed.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1654636`
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