A packing problem its application to Bose's families

We propose and study the following problem: given X ⊂ Z n , construct a maximum packing of dev X (the development of X ), i.e., a maximum set of pairwise disjoint translates of X . Such a packing is optimal when its size reaches the upper bound $\lfloor {x \over {|X|}} \rfloor$ . In particular, it is perfect when its size is exactly equal to ${n \over {|X|}}$ i.e. when it is a partition of Z n . We apply the above problem for constructing Bose's families . A ( q, k ) Bose's family (BF) is a nonempty family F of subsets of the field GF(q) such that: (i) each member of F is a coset of the k th 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 $(GF(q), {\cal B} :\ =(X+g | (X,g)\in {\cal F}\times GF(q))$ , is a semilinear space . A ( q, k )‐BF is optimal when its size reaches the upper bound $\lfloor {{q-1} \over {k(k-1)}} \rfloor$ . In particular, it is perfect when its size is exactly equal to ${{q-1} \over {k(k-1)}}$ ; in this case the ( q, 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 dev X , where X is a suitable $\lfloor {k \over 2} \rfloor$ ‐subset of Z n , $n={{q-1}\over {2k}}$ for k odd, $n={{q-1}\over {2(k-1)}}$ for k even. © 1996 John Wiley & Sons, Inc.