Partitions of $Z_2^n $

Tannenbaum, Peter

doi:10.1137/0604004

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…In [11], Tannenbaum settles this question for G = Z n 2 and n > 1. We note that the partition problem relates to the study of translation planes and to the problem of finding optimal partial spreads and has applications to byte error control codes.…”

Section: Lemma 13 (Bumentioning

confidence: 98%

On vector space partitions and uniformly resolvable designs

Blinco

El-Zanati

Seelinger

et al. 2008

Des. Codes Cryptogr.

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Let V n (q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of V n (q) is a partition of V n (q) if every nonzero vector in V n (q) is contained in exactly one subspace in P. A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V n (q) containing a i subspaces of dimension n i for 1 ≤ i ≤ k induces a uniformly resolvable design on q n points with a i parallel classes with block size q n i , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph K q n into a i K q n i -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q n points where corresponding partitions of V n (q) do not exist.

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Section: Lemma 13 (Bumentioning

confidence: 98%

On vector space partitions and uniformly resolvable designs

Blinco

El-Zanati

Seelinger

et al. 2008

Des. Codes Cryptogr.

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“…Note that Equations (11)(12)(13)(14) use exactly 28 of the 32 sets mentioned above, as shown in Table I. These sets form 28 subspaces of dimension 3 (when we include 0 in each of them).…”

Section: A Notation and Generalitiesmentioning

confidence: 98%

“…This question was settled by Tannenbaum [11] for G = Z n 2 and n > 1. We note that the partition problem has applications in combinatorial designs.…”

Section: Lemma 12 (Bu [2]) Let N D Be Integers Such That 1 ≤ D ≤ Nmentioning

confidence: 98%

Partitions of finite vector spaces into subspaces

El-Zanati

Seelinger

Sissokho

et al. 2007

J of Combinatorial Designs

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Let V n (q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of V n (q) is a partition of V n (q) if every nonzero element of V n (q) is contained in exactly one element of P. Suppose there exists a partition of V n (q) into x i subspaces of dimension n i , 1 ≤ i ≤ k. Then x 1 , . . . , x k satisfy the Diophantine equation(q n i − 1)x i = q n − 1. However, not every solution of the Diophantine equation corresponds to a partition of V n (q). In this article, we show that there exists a partition of V n (2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2 n − 1 and y = 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V n (q) induce uniformly resolvable designs on q n points.

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Partitions of $Z_2^n $

Cited by 2 publications

References 7 publications

On vector space partitions and uniformly resolvable designs

On vector space partitions and uniformly resolvable designs

Partitions of finite vector spaces into subspaces

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