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Subspace packings: constructions and bounds

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Abstract

Grassmannian \({{{\mathcal {G}}}}_q(n,k)\) is the set of all k-dimensional subspaces of the vector space \({\mathbb {F}}_q^n\). Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are q-analogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian \({{{\mathcal {G}}}}_q(n,k)\) also form a family of q-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of q-analogs of block designs which was not considered before. This family called subspace packings is the q-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing t-\((n,k,\lambda )_q\) is a set \({\mathbb {S}}\) of k-subspaces from \({{{\mathcal {G}}}}_q(n,k)\) such that each t-subspace of \({{{\mathcal {G}}}}_q(n,t)\) is contained in at most \(\lambda \) elements of \({\mathbb {S}}\). The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.

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Notes

  1. Using the methods of Sect. 3.2, we can consider the corresponding multiset \({{{\mathcal {P}}}}\) of points, which has cardinality 181 and is \(2^3\)-divisible. Its 3-complement \({\overline{{{{\mathcal {P}}}}}}\) is also 8-divisible and has cardinality 8, which leaves an 8-fold point as the unique possibility for \({\overline{{{{\mathcal {P}}}}}}\). Due to \(\lambda =3<8\), this is impossible in our situation. We remark that the Johnson bound for points, see Proposition 2, gives \({{{\mathcal {A}}}}^r_2(7,5,1;3)\le 12\), while its improvement based on the methods of Sect. 3.2, see Proposition 5, gives \({{{\mathcal {A}}}}^r_2(7,5,1;3)\le 11\).

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Acknowledgements

The third author would like to thank TÜBİTAK BİDEB for the financial support under programs 2211 and 2214/A. The third author also thanks the Computer Science Department at the Technion for their warm-hearted hospitality and support during his stay at the university between November 2017 and May 2018.

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Authors and Affiliations

  1. Computer Science Department, Technion, 3200003, Haifa, Israel

    Tuvi Etzion

  2. University of Bayreuth, Bayreuth, Germany

    Sascha Kurz

  3. TÜBİTAK BİLGEM UEKAE, Gebze, Turkey

    Kamil Otal

  4. Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

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  1. Tuvi Etzion
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Correspondence to Ferruh Özbudak.

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Appendix: Tables

Appendix: Tables

In this section we collect some numerical results on \({{{\mathcal {A}}}}_q(n,k,t;\lambda )\), i.e., the tightest lower and upper bounds known to us. We will mainly focus on the binary case \(q=2\) and small values of \(\lambda \) and give just a few tables for \(q=3\). We only provide results for \(\lambda >1\) and refer the interested reader to http://subspacecodes.uni-bayreuth.de [39] for \(\lambda =1\). In order to point to the origin of the bound or an exact formula we use the following abbreviations:

Table 1 Bounds for \({{{\mathcal {A}}}}_2(3,k,t;2)\)
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  • \(^a\): Bounds for arcs, see e.g. [3] and the end of Sect. 3.3.

  • \(^b\): Take all subspaces, see Lemma 2.

  • \(^c\): All subspaces not containing a point, see Proposition 11.

  • \(^g\): Constructions for \(q-GDDs\), a q-analog of group divisible designs, see [11].

  • \(^h\): Restriction to a hyperplane, see Proposition 4.

  • \(^i\): Intersection arguments, see Lemma 8, Propositions 10, and 6.

  • \(^{j}\): Improved Johnson bound for points, see Proposition 5.

  • \(^{k}\): Known results for packing designs, see e.g. [8].

  • \(^l\): Integer linear programming formulations.

  • \(^p\): Existence of parallel packings, see Theorem 4 in connection with the literature on large sets, and Proposition 9 in connection with the literature of (partial) parallelisms.

  • \(^q\): The quadratic upper bound from Proposition 7 based on the second-order Bonferroni Inequality.

  • \(^t\): Integer linear programming formulations with prescribed automorphisms.

  • \(^x\): Generalized linkage construction, see Theorem 3 and Corollary 2.

We remark that \({{{\mathcal {A}}}}_2(6,3,2;4)\ge 360\), which was obtained in the context of q-GDDs [11], was also obtained in [21]. The upper bound for \({{{\mathcal {A}}}}_2(6,4,2;2)\), based on integer linear programming, need a more detailed explanation, see Proposition 8, which is marked by a \(\star \) in the corresponding table. For upper bounds marked by \(^i\) we refer to the discussion directly after Proposition 6 for the details (See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 and 27.

Table 2 Bounds for \({{{\mathcal {A}}}}_2(4,k,t;2)\)
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Table 3 Bounds for \({{{\mathcal {A}}}}_2(5,k,t;2)\)
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Table 4 Bounds for \({{{\mathcal {A}}}}_2(6,k,t;2)\)
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Table 5 Bounds for \({{{\mathcal {A}}}}_2(7,k,t;2)\)
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Table 6 Bounds for \({{{\mathcal {A}}}}_2(8,k,t;2)\)
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Table 7 Bounds for \({{{\mathcal {A}}}}_2(3,k,t;3)\)
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Table 8 Bounds for \({{{\mathcal {A}}}}_2(4,k,t;3)\)
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Table 9 Bounds for \({{{\mathcal {A}}}}_2(5,k,t;3)\)
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Table 10 Bounds for \({{{\mathcal {A}}}}_2(6,k,t;3)\)
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Table 11 Bounds for \({{{\mathcal {A}}}}_2(7,k,t;3)\)
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Table 12 Bounds for \({{{\mathcal {A}}}}_2(8,k,t;3)\)
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Table 13 Bounds for \({{{\mathcal {A}}}}_2(3,k,t;4)\)
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Table 14 Bounds for \({{{\mathcal {A}}}}_2(4,k,t;4)\)
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Table 15 Bounds for \({{{\mathcal {A}}}}_2(5,k,t;4)\)
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Table 16 Bounds for \({{{\mathcal {A}}}}_2(6,k,t;4)\)
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Table 17 Bounds for \({{{\mathcal {A}}}}_2(7,k,t;4)\)
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Table 18 Bounds for \({{{\mathcal {A}}}}_2(8,k,t;4)\)
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Table 19 Bounds for \({{{\mathcal {A}}}}_3(3,k,t;2)\)
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Table 20 Bounds for \({{{\mathcal {A}}}}_3(4,k,t;2)\)
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Table 21 Bounds for \({{{\mathcal {A}}}}_3(5,k,t;2)\)
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Table 22 Bounds for \({{{\mathcal {A}}}}_3(3,k,t;3)\)
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Table 23 Bounds for \({{{\mathcal {A}}}}_3(4,k,t;3)\)
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Table 24 Bounds for \({{{\mathcal {A}}}}_3(5,k,t;3)\)
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Table 25 Bounds for \({{{\mathcal {A}}}}_3(3,k,t;4)\)
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Table 26 Bounds for \({{{\mathcal {A}}}}_3(4,k,t;4)\)
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Table 27 Bounds for \({{{\mathcal {A}}}}_3(5,k,t;4)\)
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Etzion, T., Kurz, S., Otal, K. et al. Subspace packings: constructions and bounds. Des. Codes Cryptogr. 88, 1781–1810 (2020). https://doi.org/10.1007/s10623-020-00732-z

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