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Partial Spreads and Vector Space Partitions

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Network Coding and Subspace Designs

Part of the book series: Signals and Communication Technology ((SCT))

Abstract

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake & Freeman on partial spreads date back to 1975 and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.

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Notes

  1. 1.

    Using the algebraic dimension has certain advantages—for example, the existence criterion \(v=tk\) for spreads (cf. Theorem 1) looks ugly when stated in terms of the geometric dimensions: \(v'=t(k'-1)+1\).

  2. 2.

    The use of distributed coding at the nodes of a packet-switched network, generally referred to as Network Coding, is described in [27, 56, 70] and elsewhere in this volume.

  3. 3.

    Except that attention is usually restricted to “one-shot subspace codes”, i.e. subsets of the alphabet, which makes no sense in the classical case.

  4. 4.

    Note that the distance between codewords of the same dimension, and hence also the minimum distance of a constant-dimension code, is an even integer.

  5. 5.

    In other words, partial spreads are just packings of the point set of a projective geometry \({\text {PG}}(v-1,\mathbb {F}_q)\) into subspaces of equal dimension.

  6. 6.

    The weaker property of complete (i.e., inclusion-maximal) partial spreads will not be considered.

  7. 7.

    Since \(\sum _{X\in \mathscr {C}}\dim (X)=\sum _ddm_d=v\), the type of \(\mathscr {C}\) can be viewed as an ordinary integer partition of v.

  8. 8.

    The subspace distance \(d_\text {S}(X,Y)\) depends not only on \(\dim (X\cap Y)\) but also on \(\dim (X)\) and \(\dim (Y)\), which are not constant in this case.

  9. 9.

    Segre in turn built to some extent on work of André, who had earlier considered the special case \(v=2k\) in his seminal paper on translation planes [1].

  10. 10.

    Alternatively, the member of \(\mathscr {S}\) containing a nonzero vector \(\mathbf {x}\) is the k-subspace \(\mathbb {F}_{q^k}\mathbf {x}\) of \(V/\mathbb {F}_q\).

  11. 11.

    This makes sense also for \(r=0\): Spreads are assigned deficiency \(\sigma =0\).

  12. 12.

    More generally, this formula holds if X and \(X'\) have the same pivot set and \(\mathbf {B},\mathbf {B}'\in \mathbb {F}_q^{k\times (v-k)}\) denote the corresponding complementary submatrices in their canonical matrices; see e.g. [63, Corollary 3].

  13. 13.

    In ring-theoretic terms, the matrices in \(M(\mathbb {F}_{q^n})\) form a maximal subfield of the ring of \(n\times n\) matrices over \(\mathbb {F}_q\).

  14. 14.

    The space \(S_0\) has been named moving subspace of \(\mathscr {S}\), since it can be freely “moved” within S without affecting the partial spread property of \(\mathscr {S}\).

  15. 15.

    The full classification, including also partial line spreads of smaller size, can be found in [26]. To our best knowledge there is only one further nontrivial parameter case, where a classification of maximal (proper) partial spreads is known, viz. the case of plane spreads in \({\text {PG}}(6,\mathbb {F}_2)\), settled in [38].

  16. 16.

    In this particular case one may also argue as follows: If \(\#\mathscr {S}=10\) then there is only one hole and the hyperplane constraint becomes \(3\alpha +(10-\alpha )+h=15\), where \(h\in \{0,1\}\). This forces \(\alpha =2\) and \(h=1\), i.e., every hyperplane should contain the hole. This is absurd, of course.

  17. 17.

    This can also written as \(\mathrm {A}_q(v,2k;k)= q^1\cdot \frac{q^{v-1}-1}{q^k-1}-q+1=\frac{q^v-q^{k+1}+q^k-1}{q^k-1}\).

  18. 18.

    The corresponding number of holes is \(q^k\).

  19. 19.

    Assuming \(1+4q^k(q^k-q^r)=1+4q^{k+r}(q^{k-r}-1)=(2z-1)^2=1+4z(z-1)\) for some integer \(z>1\) implies \(q^{k+r}\mid z\) or \(q^{k+r}\mid z-1\), so that \(z\ge q^{k+r}\), which is impossible for \((k,r)\ne (1,0)\). Thus, \(2\theta \notin \mathbb {Z}\), so that \(\theta \notin \mathbb {Z}\) and \(\lfloor \theta \rfloor +1=\lceil \theta \rceil \).

  20. 20.

    Thus in all these cases \(\sigma =2^2-1\) and the partial spreads of Theorem 2 are maximal. This notably differs from the case \(k=3\).

  21. 21.

    The upper bound can be sharpened to 132, as we will see later.

  22. 22.

    This corresponds again to the upper bound \(\sigma =q^r-1\).

  23. 23.

    This is also true for \(v=1\), where \(\mathscr {C}=\emptyset ,\mathscr {P}\) exhausts all possibilities.

  24. 24.

    The general (multiset) version of (3.4) has an additional summand of \(q^{v-2}\cdot \sum _{P\in \mathscr {P}}\left( {\begin{array}{c}\mathscr {C}(P)\\ 2\end{array}}\right) \) on the right-hand side, accounting for the fact that “pairs of equal points” are contained in \(\genfrac[]{0.0pt}{}{v-1}{1}_{q}\) hyperplanes.

  25. 25.

    This definition can be extended to multisets \(\mathscr {C}\) by defining the multiplicity of Y / X in \(\mathscr {C}/X\) as the sum of the multiplicities in \(\mathscr {C}\) of all points in \(Y\setminus X\).

  26. 26.

    If \(C\leftrightarrow \mathscr {C}\) then the multisets \(\mathscr {C}/P\), \(P\in \mathscr {P}\), are associated to the (\(v-1\))-dimensional subcodes \(D\subset C\), and the n points \(P\in \mathscr {C}\) correspond to the n subcodes D of effective length \(n-1\) (“D is C shortened at P”). This correspondence between points and subcodes extends to a correlation between \({\text {PG}}(v-1,\mathbb {F}_q)\) and \({\text {PG}}(C/\mathbb {F}_q)\), which includes the familiar correspondence between hyperplanes and codewords as a special case; see [18, 65] for details.

  27. 27.

    It is not required that \(\mathscr {C}\) is spanning; if it is not then (iii) sharpens to “\(\varDelta \) is a divisor of \(q^{\dim \langle \mathscr {C}\rangle -2}\)”.

  28. 28.

    Readers may have noticed that, curiously, the 3rd standard equation (which characterizes projective codes) was not used at all in the proof.

  29. 29.

    By this we mean that \(\varDelta \) is the largest divisor in the sense of Definition 1 or Theorem 7.

  30. 30.

    If \(t=\lfloor r\rfloor \) and \(r'\in \{0,1,\dots ,e-1\}\) is defined by \(r=t+ r'/e\), then the union of \(p^{r'}\) parallel affine subspaces of dimension \(t+1\) has this property.

  31. 31.

    This follows, e.g., by applying the Griesmer bound to the associated linear code, which has minimum distance \(\ge q^{k-1}\) and dimension \(\ge k\).

  32. 32.

    Just recall that the length of any doubly-even self-dual binary code must be a multiple of 8.

  33. 33.

    Adding \(20=5\cdot 4\), which accounts for the 5 pairs of equal points in the code, to the right-hand side “corrects” the third equation.

  34. 34.

    This assumption is necessary for the relation \(A_i=(q-1)a_{n-i}\) to hold.

  35. 35.

    Typically, the \(A_i^\perp \) are removed from the formulation using the explicit formulas based on the Krawtchouk polynomials, which may of course also be done automatically in the preprocessing step of a customary linear programming solver.

  36. 36.

    The use of a special polynomial, like we will do, is well known in the context of the linear programming method, see e.g. [9, Sect. 18.1].

  37. 37.

    This result is not new at all. In [7] Beutelspacher used such an average argument in his upper bound on the size of partial spreads. Recently, Năstase and Sissokho used it in [59, Lemma 9]. In coding theory it is well known in the context of the Griesmer bound. One may also interpret it as an easy implication of the first two MacWilliams identities, see Lemma 20 and Corollary 9.

  38. 38.

    Another parametrization for y is given by \(y=qb'-b\), where \(b'\in \mathbb {Z}\) with \(b'\ge \frac{b}{q}\) and \(b'\equiv b\pmod {q^{r+1}}\), so that \(y\in \mathbb {N}_0\). Due to \(b'=\frac{b+y}{q}\), y is minimal if and only if \(b'\) is minimal.

  39. 39.

    The proof shows that the second assertion of the Corollary is true for all \((v-j)\)-subspaces U.

  40. 40.

    If we relax \(\ge 0\)-inequalities by adding some auxiliary variable on the left hand side and the minimization of this variable, we can remove the infeasibility, so that we apply the duality theorem of linear programming. Then, the mentioned multipliers for the inequalities are given as the solution values of the dual problem.

  41. 41.

    For more general non-existence results of vector space partitions see e.g. [29, Theorem 1] and the related literature. Actually, we do not need the assumption of an underlying vector space partition of the mentioned type. The result is generally true for \(q^{s-1}\)-divisible codes, since the parameter x is just a nice technical short-cut to ease the notation.

  42. 42.

    Solving \(\frac{\partial \tau _q(c,\varDelta ,m)}{\partial m}=0\), i.e., minimizing \(\tau _q(c,\varDelta ,m)\), yields \(m=i(q-1)-(x-1)+\frac{1}{2}+\frac{x-1}{q^y}\). For \(y\ge r\) we can assume \(x-1<q^y\) due to Theorem 2, so that up-rounding yields the optimum integer choice. For \(y<r\) the interval \(\left[ \lambda +\frac{1}{2}- \frac{1}{2}\theta (i),\lambda +\frac{1}{2}+ \frac{1}{2}\theta (i)\right] \) may contain no integer.

  43. 43.

    For \(2\le q\le 9, 1\le v,k\le 100\) the bounds of [58, Theorem 6,7] are covered by Theorem 10 and Corollary 7. In many cases the bounds coincide.

  44. 44.

    The generalization to more than one “removed” subspace is also quite obvious and expresses the divisibility properties of optimal linear codes of type BV in the projective case [4, 35, 50].

  45. 45.

    If not both \(\mathscr {C}_1\) and \(\mathscr {C}_2\) are subspaces, then disjoint embeddings into a geometry \({\text {PG}}(v-1,\mathbb {F}_q)\) with \(v<\dim \langle \mathscr {C}_1\rangle +\dim \langle \mathscr {C}_2\rangle \) exist as well.

  46. 46.

    Note that \(\gcd \!\left( \genfrac[]{0.0pt}{}{r+1}{1}_{q},q^{r+1}\right) =1\) and recall the solution of the ordinary Frobenius Coin Problem.

  47. 47.

    Our sunflowers need not have constant dimension, however.

  48. 48.

    Note that \(m(q-1)\equiv 0\pmod {q^{r+1}}\) is equivalent to \(m\equiv 0\pmod {q^{r+1}}\).

  49. 49.

    In the case \(q=p\), and in general for codes of type BV, such codes are even \(q^e\)-divisible, where \(q^e\) is the largest power of p dividing the minimum distance [67, Theorem 1 and Proposition 2].

  50. 50.

    The three examples are realized in dimensions 6, 7 and 8, respectively. Alternative solutions for \(n\in \{15,16\}\), having smaller ambient space dimensions, are the [15, 4, 8] simplex code and the [16, 5, 8] first-order Reed-Muller code.

  51. 51.

    These examples can be realized in \(\mathbb {F}_2^6\) for \(n\in \{17,18\}\) and in \(\mathbb {F}_2^7\) for \(n\in \{19,20\}\).

  52. 52.

    It might look tempting to construct a projective 8-divisible binary code of length 50 by shortening such a code C of length 51. However, this does not work: By Lemma 24, C is the concatenation of an ovoid in \({\text {PG}}(3,\mathbb {F}_4)\) with the binary [3, 2] simplex code. By construction, the corresponding 8-divisible point set \(\mathscr {C}\) is the disjoint union of 17 lines. In particular, each point of \(\mathscr {C}\) is contained in a line in \(\mathscr {C}\). Consequently, shortening C in any coordinate never gives a projective code.

  53. 53.

    Consequently, for all \(t\ge 2\) the upper bound for \(\mathrm {A}_2(4t+3,8;4)\) is tightened by one; cf. Lemma 4.

  54. 54.

    See http://www.rlmiller.org/de_codes and [19] for the classification of, possibly non-projective, doubly-even codes over \(\mathbb {F}_2^v\).

  55. 55.

    Alternatively, the 4th MacWilliams identity yields \(64-2^{k-2}=2^{k-3}\cdot A_3^\perp \) and hence \(k\le 8\).

  56. 56.

    624 non-isomorphic examples can be found at http://subspacecodes.uni-bayreuth.de [33].

  57. 57.

    In a forthcoming paper we classify the 2612 non-isomorphic partial 3-spreads of cardinality 34 in \(\mathbb {F}_2^8\) that admit an automorphism group of order exactly 8, which is possible for \(\mathscr {H}_6\) only, and show that the automorphism groups of all other examples have order at most 4.

  58. 58.

    Since punctured affine solids are associated to the \([7,4,3]_2\) Hamming code, we may also think of \(\mathscr {H}_8\) as consisting of the 2-fold repetition of the [5, 4, 2]-code and the Hamming code “glued together” in Q. In fact the doubly-even \([17,8,4]_2\) code associated with \(\mathscr {H}_8\) is the code \(\overline{I}_{17}^{(3)}\) in [61, p. 234]. The glueing construction is visible in the generator matrix.

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Acknowledgements

The authors would like to acknowledge the financial support provided by COST – European Cooperation in Science and Technology. The first author was also supported by the National Natural Science Foundation of China under Grant 61571006. The third author was supported in part by the grant KU 2430/3-1 – Integer Linear Programming Models for Subspace Codes and Finite Geometry from the German Research Foundation.

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  1. Zhejiang University, Hangzhou, 310027, China

    Thomas Honold

  2. University of Bayreuth, 95440, Bayreuth, Germany

    Michael Kiermaier & Sascha Kurz

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  1. Thomas Honold
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  2. Michael Kiermaier
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  3. Sascha Kurz
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Correspondence to Sascha Kurz .

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  1. Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland

    Marcus Greferath

  2. Faculty of Electrical Engineering and Computing, Department of Applied Mathematics, University of Zagreb, Zagreb, Croatia

    Mario Osvin Pavčević

  3. Yahoo Research, Haifa, Israel

    Natalia Silberstein

  4. Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona, Spain

    María Ángeles Vázquez-Castro

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Honold, T., Kiermaier, M., Kurz, S. (2018). Partial Spreads and Vector Space Partitions. In: Greferath, M., Pavčević, M., Silberstein, N., Vázquez-Castro, M. (eds) Network Coding and Subspace Designs. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-70293-3_7

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  • DOI: https://doi.org/10.1007/978-3-319-70293-3_7

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