The Lattice of Cyclic Flats of a Matroid

Bonin, Joseph E.; Mier, Anna de

doi:10.1007/s00026-008-0344-3

Cited by 46 publications

(64 citation statements)

References 13 publications

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“…Theorem 2.2 (see [18] Th. 3.2): Let Z be a collection of subsets of a set E and let ρ be a function ρ : Z → Z.…”

Section: B Basic Properties Of the Lattice Of Cyclic Flatsmentioning

confidence: 95%

“…We remark that, by the use of basic matroid theory and some properties of cyclic flats given in [18], we can obtain the following characterization of coA Z of an (n, k, d, r)-matroid…”

Section: Equation (3) Implies Thatmentioning

confidence: 97%

“…The poset (Z, ⊆) of a matroid is a lattice, and the following basic properties of (Z, ⊆) can for example be found in [18].…”

Section: B Basic Properties Of the Lattice Of Cyclic Flatsmentioning

confidence: 99%

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Almost affine locally repairable codes and matroid theory

Westerbäck

Ernvall

Hollanti

2014

2014 IEEE Information Theory Workshop (ITW 2014)

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In this paper we provide a link between matroid theory and locally repairable codes (LRCs) that are almost affine. The parameters (n, k, d, r) of LRCs are generalized to matroids. A bound on the parameters (n, k, d, r), similar to the bound in [P. Gopalan et al., "On the locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs, is given for matroids. We prove that the given bound is not tight for a certain class of parameters, which implies a non-existence result for a certain class of optimal locally repairable almost affine codes. Constructions of optimal LRCs over small finite fields were stated as an open problem in [I. Tamo et al., "Optimal locally repairable codes and connections to matroid theory", 2013 IEEE ISIT]. In this paper optimal LRCs which do not require a large field are constructed for certain classes of parameters.

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“…Theorem 2.2 (see [18] Th. 3.2): Let Z be a collection of subsets of a set E and let ρ be a function ρ : Z → Z.…”

Section: B Basic Properties Of the Lattice Of Cyclic Flatsmentioning

confidence: 95%

“…We remark that, by the use of basic matroid theory and some properties of cyclic flats given in [18], we can obtain the following characterization of coA Z of an (n, k, d, r)-matroid…”

Section: Equation (3) Implies Thatmentioning

confidence: 97%

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Almost affine locally repairable codes and matroid theory

Westerbäck

Ernvall

Hollanti

2014

2014 IEEE Information Theory Workshop (ITW 2014)

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“…Hence Z obeys (Z3) for all k ≥ 4, so M is a matroid. As noted in [3], its circuits are the minimal subsets S of E such that Z contains an element Z containing S with |S| = r(Z) + 1.…”

Section: Preliminariesmentioning

confidence: 99%

Generalized laminar matroids

Fife

Oxley

2019

European Journal of Combinatorics

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Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid M is Hamiltonian if it has a spanning circuit. A matroid M is nested if and only if its Hamiltonian flats form a chain under inclusion; M is laminar if and only if, for every 1-element independent set X, the Hamiltonian flats of M containing X form a chain under inclusion.We generalize these notions to define the classes of k-closure-laminar and k-laminar matroids. This paper focuses on structural properties of these classes noting that, while the second class is always minorclosed, the first is if and only if k ≤ 3. The main results are excluded-minor characterizations for the classes of 2-laminar and 2-closure-laminar matroids.

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“…Let M = (ρ, E) be a matroid. The closure operator cl : 2 E → 2 E and cyclic operator cyc : 2 E → 2 E are defined by It is easy to verify, as in [2], that the closure operator induces flatness and preserves cyclicity, and that the cyclic operator induces cyclicity and preserves flatness. Thus we can write cl :…”

Section: Fundamentals On Cyclic Flatsmentioning

confidence: 99%

On Binary Matroid Minors and Applications to Data Storage over Small Fields

Grezet

Freij-Hollanti

Westerbäck

et al. 2017

Coding Theory and Applications

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Locally repairable codes for distributed storage systems have gained a lot of interest recently, and various constructions can be found in the literature. However, most of the constructions result in either large field sizes and hence too high computational complexity for practical implementation, or in low rates translating into waste of the available storage space. In this paper we address this issue by developing theory towards code existence and design over a given field. This is done via exploiting recently established connections between linear locally repairable codes and matroids, and using matroid-theoretic characterisations of linearity over small fields. In particular, nonexistence can be shown by finding certain forbidden uniform minors within the lattice of cyclic flats. It is shown that the lattice of cyclic flats of binary matroids have additional structure that significantly restricts the possible locality properties of F2-linear storage codes. Moreover, a collection of criteria for detecting uniform minors from the lattice of cyclic flats of a given matroid is given, which is interesting in its own right.The parameters (n, k, d, r, δ) can immediately be defined and studied for matroids in general, as in [18,20]. Matroid fundamentalsMatroids were first introduced by Whitney in 1935, to capture and generalise the notion of linear dependence in purely combinatorial terms.

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The Lattice of Cyclic Flats of a Matroid

Cited by 46 publications

References 13 publications

Almost affine locally repairable codes and matroid theory

Almost affine locally repairable codes and matroid theory

Generalized laminar matroids

On Binary Matroid Minors and Applications to Data Storage over Small Fields

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