Abstract
AbstractLet be a fixed prime power and let denote a vector space of dimension over the Galois field with elements. A subspace partition (also called “vector space partition”) of is a collection of subspaces of with the property that every nonzero element of appears in exactly one of these subspaces. Given positive integers such that , we say a subspace partition of has type if it is composed of subspaces of dimension and subspaces of dimension . Let . In this paper, we prove that if divides , then one can (algebraically) construct every possible subspace partition of of type whenever , where and . Our construction allows us to sequentially reconfigure batches of subspaces of dimension into batches of subspaces of dimension . In particular, this accounts for all numerically allowed subspace partition types of under some additional conditions, for example, when .
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