Higgledy-piggledy subspaces and uniform subspace designs

Fancsali, Szabolcs L.; Sziklai, Péter

doi:10.1007/s10623-016-0189-4

Cited by 7 publications

(7 citation statements)

References 8 publications

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“…We now prove the new characterization of strong blocking sets based on affine blocking sets, outlined in Lemma 1.2. First, we define a generalization of strong blocking sets, which also appears in [27,Definition 2] under the name of generator sets. Definition 2.21.…”

Section: 2mentioning

confidence: 99%

“…If the strong (𝑠 − 1)-blocking set has size 𝑚, then the corresponding affine 𝑠-blocking set has size (𝑞 − 1)𝑚 + 1. For example, if 𝑞 > 𝑠(𝑘 − 𝑠), there exists an explicit construction of strong (𝑠 − 1)blocking sets of size at most (𝑠(𝑘 − 𝑠) + 1)(𝑞 𝑠 − 1)∕(𝑞 − 1) in PG(𝑘 − 1, 𝑞) [27,33], which we can use to give an explicit construction of affine 𝑠-blocking sets of size at most (𝑞 𝑠 − 1)𝑠(𝑘 − 𝑠) + 𝑞 𝑠 . While this is a good explicit construction, it requires the field to be large with respect to 𝑘 and 𝑠.…”

Section: Explicit Constructionsmentioning

confidence: 99%

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Blocking sets, minimal codes and trifferent codes

Bishnoi,

D'haeseleer,

Gijswijt

et al. 2024

Journal of London Math Soc

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We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension‐2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on a strong blocking set. Our main motivation for these new bounds is their application to trifferent codes, which are sets of ternary codes of length with the property that for any three distinct codewords there is a coordinate where they all have distinct values. Over the finite field , we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length has size at most , improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length and size at least , thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension‐2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to , we obtain linear trifferent codes of size at least , improving the current best explicit construction that has size .

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Section: 2mentioning

confidence: 99%

Section: Explicit Constructionsmentioning

confidence: 99%

Blocking sets, minimal codes and trifferent codes

Bishnoi,

D'haeseleer,

Gijswijt

et al. 2024

Journal of London Math Soc

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“…We now prove the new characterization of strong blocking sets based on affine blocking sets, outlined in Lemma 1.2. First, we define a generalization of strong blocking sets, which also appears in [23,Definition 2] under the name of generator sets. Definition 2.21.…”

Section: Strong Blocking Setsmentioning

confidence: 99%

“…If the strong (s − 1)-blocking set has size m, then the corresponding affine s-blocking set has size (q − 1)m + 1. For example, if q > s(k − s), there exists an explicit construction of strong (s − 1)-blocking sets of size at most (s(k − s) + 1)(q s − 1)/(q − 1) in PG(k − 1, q) [23,29], which we can use to give an explicit construction of affine s-blocking sets of size at most (q s − 1)s(k − s) + q s . While this is a good explicit construction, it requires the field to be large with respect to k and s. We will focus on fixed q, s, and large k.…”

Section: Explicit Constructionsmentioning

confidence: 99%

Blocking sets, minimal codes and trifferent codes

Bishnoi¹,

D'haeseleer²,

Gijswijt³

et al. 2023

Preprint

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We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set.Over the finite field F 3 , we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length n has size at most 3 n/4.55 , improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length n and size at least 1 3 (9/5) n/4 , thus (asymptotically) matching the best lower bound on trifferent codes.We also give explicit constructions of affine blocking sets with respect to codimension-2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to F 3 , we obtain linear trifferent codes of size at least 3 7n/240 , improving the current best explicit construction that has size 3 n/112 .

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“…The name arises from cutting blocking sets, recently introduced by Bonini and Borello in [21], with the aim of constructing minimal codes. This structure has been also studied before the paper [21], under other names such as strong blocking sets and generator sets, in connection with saturating sets and covering codes by Davydov, Giulietti, Marcugini and Pambianco [30] and higgledy-piggledy line arrangements by Fancsali and Sziklai [38] (see also [52] and [39]). In this section we propose a generalization of this notion to subspace designs which turns out to be connected with minimal sum-rank metric codes, as we will see later.…”

Section: Cutting Design and Minimal Sum-rank Metric Codesmentioning

confidence: 99%

On subspace designs

Santonastaso¹,

Zullo²

2022

Preprint

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The aim of this paper is to investigate the theory of subspace designs, which have been originally introduced by Guruswami and Xing in 2013 to give the first construction of positive rate rank-metric codes list decodable beyond half the distance. However, sets of subspaces with special pattern of intersections with other subspaces have been already studied, such as spreads, generalised arcs and caps, and Cameron-Liebler sets. In this paper we provide bounds involving the dimension of the subspaces forming a subspace design and the parameters of the ambient space, showing constructions satisfying the equality in such bounds. Then we also introduce two dualities relations among them. Among subspace designs, those that we call s-designs are central in this paper, as they generalize the notion of s-scattered subspace to subspace design and for their special properties. Indeed, we prove that, for certain values of s, they correspond to the optimal subspace designs, that is those subspace designs that are associated with linear maximum sum-rank metric codes. Special attention has been paid for the case s = 1 for which we provide several examples, yielding surprising to families of two-intersection sets with respect to hyperplanes (and hence two-weight linear codes). Moreover, s-designs can be used to construct explicit lossless dimension expanders (a linear-algebraic analogue of expander graphs), without any restriction on the order of the field. Another class of subspace designs we study is those of cutting designs, since they extend the notion of cutting blocking set recently introduced by Bonini and Borello. These designs turn out to be very interesting as they in one-to-one correspondence with minimal sum-rank metric codes. The latter codes have been introduced in this paper and they naturally extend the notions of minimal codes in both Hamming and rank metrics.

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Higgledy-piggledy subspaces and uniform subspace designs

Cited by 7 publications

References 8 publications

Blocking sets, minimal codes and trifferent codes

Blocking sets, minimal codes and trifferent codes

Blocking sets, minimal codes and trifferent codes

On subspace designs

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