The maximum size of a partial 3-spread in a finite vector space over GF(2)

El-Zanati, Saad I.; Jordon, Heather; Seelinger, George F.; Sissokho, Papa A.; Spence, Lawrence E.

doi:10.1007/s10623-009-9311-1

Cited by 39 publications

(53 citation statements)

References 18 publications

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“…Moreover, we can obtain binary linear LRCs with parameters [20,8,8; 3] 2 and [24, 11, 8; 3] 2 by using the method in Construction IV.18. All of these examples are optimal with respect to bound (5).…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…(See Figure 1 below for an illustration.) s l n k 4 [3,5] 3l 2l − 4 5 [6,9] 3l 2l − 5 6 [11,21] 3l 2l − 6 7 [22,41] 3l 2l − 7 2) The second class of optimal binary LRCs:…”

Section: A K-optimal Constructions For D =mentioning

confidence: 99%

“…Example IV.12. Taking t = 3, we obtain a binary linear code with parameters [11,4,5]. Let {W 1 , W 2 , · · · , W 129 } be a 7-spread of F 14 2 .…”

Section: A K-optimal Constructions For D =mentioning

confidence: 99%

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Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

Ma¹,

Ge²

2019

SIAM J. Discrete Math.

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In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary LRCs are of interest in practice.In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension k of such codes, which can be served as a generalization of the bounds given in [11], [36], [37]. We also give several new constructions of binary LRCs with minimum distance d = 6 based on weakly independent sets and partial spreads, which are optimal with respect to our newly obtained bound. In particular, for locality r ∈ {2, 3} and minimum distance d = 6, we obtain the desired optimal binary linear LRCs with disjoint repair groups for almost all parameters.which is also called as Singleton-like bound since it degenerates to classical Singleton bound when r = k. Later, in [7], [19], the bound (1) is generalized to vector codes and nonlinear codes. Although it certainly holds for all LRCs, it is not tight in many cases. The tightness of the bound (1) is studied in [28], [34].We say an LRC is d-optimal if it satisfies bound (1) with equality for given n, k and r. For the case (r + 1)|n, d-optimal LRCs are constructed explicitly in [32] and [25] by using Reed-Solomon codes and Gabidulin codes respectively. However, both constructions are built over a finite field whose size is an exponential function of the code length n. In [30], for the same case (r + 1)|n, the authors construct a d-optimal code over a finite field of size sightly greater than n by using "good" polynomials. This construction can be extended to the case (r + 1) ∤ n with the minimum distance d ≥ n − k − ⌈ k r ⌉ + 1 which

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Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…(See Figure 1 below for an illustration.) s l n k 4 [3,5] 3l 2l − 4 5 [6,9] 3l 2l − 5 6 [11,21] 3l 2l − 6 7 [22,41] 3l 2l − 7 2) The second class of optimal binary LRCs:…”

Section: A K-optimal Constructions For D =mentioning

confidence: 99%

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Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

Ma¹,

Ge²

2019

SIAM J. Discrete Math.

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“…The determination of A 2 (v, 6; 3) for v ≡ 2 (mod 3) was achieved more than 30 years later in [14] and continued to A 2 (v, 2k; k) for v ≡ 2 (mod k) and arbitrary k in [34]. Besides the parameters of A 2 (8 + 3l, 6; 3), for l ≥ 0, see [14] for an example showing A 2 (8, 6; 3) ≥ 34, no partial spreads exceeding the lower bound from Theorem 12 are known.…”

Section: Upper Bounds For Partial Spreadsmentioning

confidence: 99%

Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

Heinlein

Kurz

2017

Coding Theory and Applications

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We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance d = 4, dimension k = 3 of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

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“…Vector space partitions have applications in design theory (in particular, uniformly resolvable designs [1]), coding theory (see [3,[12][13][14]), and orthogonal arrays (see [4,7]). The study of vector space partitions of V (n, q) for small n and q is important in providing a rich set of examples and in supporting more general results.…”

Section: Introductionmentioning

confidence: 99%

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

El-Zanati

Heden

Seelinger

et al. 2010

J of Combinatorial Designs

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Let V = V(n,q) denote the vector space of dimension n over GF(q). A set of subspaces of V is called a partition of V if every nonzero vector in V is contained in exactly one subspace of V. Given a partition P of V with exactly a i subspaces of dimension i for 1 ≤ i ≤ n, we have n i=1 a i (q i −1) = q n −1, and we call the n-tuple (a n , a n-1 ,...,a 1 ) the type of P. In this article we identify all 8-tuples (a 8 , a 7 ,...,a 2 , 0) that are the types of partitions of V(8,2). q

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The maximum size of a partial 3-spread in a finite vector space over GF(2)

Cited by 39 publications

References 18 publications

Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

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

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