Lossless Dimension Expanders Via Linearized Polynomials and Subspace Designs

Guruswami, Venkatesan; Resch, Nicolas; Xing, Chaoping

doi:10.1007/s00493-020-4360-1

Cited by 8 publications

(12 citation statements)

References 28 publications

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“…Finally, let us mention that higgledy-piggledy lines and subspaces are also related to uniform subspace designs. For the definition and coding theoretic applications of subspace designs, we refer to the works of Guruswami and Kopparty [24], Guruswami, Resch, and Xing [25] and the references therein, whereas the relation of uniform subspace designs and higgledy-piggledy subspaces can be found in the work of Fancsali and Sziklai [22].…”

Section: Explicit Results For Small Nmentioning

confidence: 99%

Short minimal codes and covering codes via strong blocking sets in projective spaces

Héger¹,

Nagy²

2021

Preprint

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Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension k over the q-element Galois field which is linear in both q and k, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. The contributions rely on geometric and probabilistic arguments.

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Section: Explicit Results For Small Nmentioning

confidence: 99%

Short minimal codes and covering codes via strong blocking sets in projective spaces

Héger¹,

Nagy²

2021

Preprint

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“…Explicit subspace designs were constructed by Guruswami and Kopparty [GK16] and also by Guruswami, Xing, and Yuan [GXY18]. They have applications to constructing explicit list-decodable codes with small list size [GX13, GWX16, KRZSW18, GRZ20] and explicit dimension expanders [FG15,GRX21]. Subspace designs were also used to prove lower bounds in communication complexity [CGS20].…”

Section: Other Related Workmentioning

confidence: 99%

Variety Evasive Subspace Families

Guo

2021

Preprint

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We introduce the problem of constructing explicit variety evasive subspace families. Given a family F of subvarieties of a projective or affine space, a collection H of projective or affine k-subspaces is (F , ǫ)evasive if for every V ∈ F, all but at most ǫ-fraction of W ∈ H intersect every irreducible component of V with (at most) the expected dimension. The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers.Using Chow forms, we construct explicit k-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine n-space. As one application, we obtain a complete derandomization of Noether's normalization lemma for varieties of bounded degree in a projective or affine n-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester-Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130).As a complement of our explicit construction, we prove a lower bound for the size of k-subspace families that are evasive for degree-d varieties in a projective n-space. When n − k = Ω(n), the lower bound is superpolynomial unless d is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.

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“…A subspace design is a collection of subspaces with the property that no low-dimensional subspace intersects with too many subspaces from the collection. Constructions of subspace designs have found several applications, such as constructing list-decodable error-correcting codes [5], rank-metric codes [4], dimensional expanders [3]. In particular, using this concept the first deterministic polynomial time construction of list-decodable codes over constant-sized large alphabets and sub-logarithmic list size achieving the optimal rate has been designed [2].…”

Section: Related Workmentioning

confidence: 99%

Almost Affinely Disjoint Subspaces

Liu,

Polyanskii,

Vorobyev

et al. 2020

Preprint

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In this work, we introduce a natural notion concerning vector finite spaces. A family of k-dimensional subspaces of F n q is called almost affinely disjoint if any (k + 1)-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the paper is the polynomial growth (in q) of the maximal cardinality of these families given the parameters k and n. For the cases k = 1 and k = 2, optimal families are constructed. For other settings, we find lower and upper bounds on the polynomial growth. Additionally, some connections with problems in coding theory are shown.

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Lossless Dimension Expanders Via Linearized Polynomials and Subspace Designs

Cited by 8 publications

References 28 publications

Short minimal codes and covering codes via strong blocking sets in projective spaces

Short minimal codes and covering codes via strong blocking sets in projective spaces

Variety Evasive Subspace Families

Almost Affinely Disjoint Subspaces

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