Partitions of the 8‐dimensional vector space over <i>GF</i>(2)

El-Zanati, Saad I.; Heden, Olof; Seelinger, George F.; Sissokho, Papa A.; Spence, Lawrence E.; Eynden, Charles Vanden

doi:10.1002/jcd.20247

Cited by 12 publications

(11 citation statements)

References 13 publications

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“…All partition types of

V (n, q)

are known in the following cases: for

q = 2

k = 2

d_{2} = 3

, and

d_{1} = 2

(El‐Zanati et al ); for

q = 2

and

n \leq 7

(El‐Zanati et al ); for

q = 2

n = 8

, and

d_{i} \geq 2

for all

1 \leq i \leq k

(El‐Zanati et al ); for

q = 2

k = 3

n \geq 9

d_{3} = n - 3

d_{2} = 3

, and

d_{1} = 2

(Heden ); and finally for

k \geq q + 1

and

d_{q + 1} = d_{q + 2} = . . . = d_{k}

(Heden ).…”

Section: Introduction and Supporting Resultsmentioning

confidence: 99%

Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

Seelinger¹,

Sissokho²,

Spence³

et al. 2012

J of Combinatorial Designs

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Let V=V(n,q) denote a vector space of dimension n over the field with q elements. A set scriptP of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in scriptP. Suppose that scriptP is a partition of V with xi subspaces of dimension di for 1≤i≤k. Then we call d1x1...dkxk the type of the partition scriptP. Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer s≥3 and for any integer n≥2s, the existence of partitions of V(n,q) across a suitable range of types sx2y guarantees the existence of partitions of V(n+js,q) of essentially all the types sx2y for any integer j≥1. We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012

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“…All partition types of

V (n, q)

are known in the following cases: for

q = 2

k = 2

d_{2} = 3

, and

d_{1} = 2

(El‐Zanati et al ); for

q = 2

and

n \leq 7

(El‐Zanati et al ); for

q = 2

n = 8

, and

d_{i} \geq 2

for all

1 \leq i \leq k

(El‐Zanati et al ); for

q = 2

k = 3

n \geq 9

d_{3} = n - 3

d_{2} = 3

, and

d_{1} = 2

(Heden ); and finally for

k \geq q + 1

and

d_{q + 1} = d_{q + 2} = . . . = d_{k}

(Heden ).…”

Section: Introduction and Supporting Resultsmentioning

confidence: 99%

Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

Seelinger¹,

Sissokho²,

Spence³

et al. 2012

J of Combinatorial Designs

Self Cite

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show abstract

“…However, there are some rare cases where the existence of a vector space partition was excluded with more involved techniques, see e.g. [12] for the exclusion of a vector space partition of type 4 13 3 6 2 6 in F 8 2 . Nevertheless, the classification of all possible cardinalities of q r -divisible sets of points is an important relaxation.…”

Section: Q R -Divisible Sets Of T-subspacesmentioning

confidence: 99%

Generalized vector space partitions

Heinlein¹,

Honold²,

Kiermaier³

et al. 2018

Preprint

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“…n different types of vector space partitions in V (n, 2) Together with Heden, El-Zanati et al [16] investigated the case n = 8 and q = 2 and vector space partitions consisting of spaces of dimension at least equal to 2. It turned out that the packing condition, dimension condition and the tail condition, with just one exception, were both necessary and sufficient in this case.…”

Section: 3mentioning

confidence: 99%

A Survey of the Different Types of Vector Space Partitions

Heden

2012

Discrete Math. Algorithm. Appl.

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A vector space partition is here a collection P of subspaces of a finite vector space V (n, q), of dimension n over a finite field with q elements, with the property that every non zero vector is contained in a unique member of P. Vector space partitions relates to finite projective planes, design theory and error correcting codes.In the first part of the talk I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the talk contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V (n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition.Finally, I will also give a few historical remarks.

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Partitions of the 8‐dimensional vector space over GF(2)

Cited by 12 publications

References 13 publications

Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

Generalized vector space partitions

A Survey of the Different Types of Vector Space Partitions

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