Finite topologies and Hamiltonian paths

Anderson, Bruce

doi:10.1016/s0095-8956(73)80008-5

Cited by 49 publications

(42 citation statements)

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“…We distinguish between three cases. A similar proof, but for a different combinatorial construct (which does not yield quasi-cyclic codes) appears in [1].…”

Section: Proposition 10mentioning

confidence: 82%

“…In the finite field with seven elements, , 1 pick , an element of multiplicative order . Pick , an element with multiplicative order .…”

Section: : Cyclic Lowest Density Mds Codes Withmentioning

confidence: 99%

“…The sets are used in [4] to construct (noncyclic) lowest density MDS codes with redundancy . The same construction, only with , provides (noncyclic) lowest density MDS codes by applying the perfect -factorization of complete graphs with vertices by Anderson [1], to the construction of [8]. Shortened versions of the noncyclic constructions of [8] and [4] are used in the proofs of the constructions of this paper, and are denoted and , respectively.…”

Section: : Cyclic Lowest Density Mds Codes Withmentioning

confidence: 99%

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Cyclic Lowest Density MDS Array Codes

Cassuto

Bruck

2009

IEEE Trans. Inform. Theory

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Abstract-Three new families of lowest density maximum-distance separable (MDS) array codes are constructed, which are cyclic or quasi-cyclic. In addition to their optimal redundancy (MDS) and optimal update complexity (lowest density), the symmetry offered by the new codes can be utilized for simplified implementation in storage applications. The proof of the code properties has an indirect structure: first MDS codes that are not cyclic are constructed, and then transformed to cyclic codes by a minimum-distance preserving transformation.

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“…We distinguish between three cases. A similar proof, but for a different combinatorial construct (which does not yield quasi-cyclic codes) appears in [1].…”

Section: Proposition 10mentioning

confidence: 82%

“…In the finite field with seven elements, , 1 pick , an element of multiplicative order . Pick , an element with multiplicative order .…”

Section: : Cyclic Lowest Density Mds Codes Withmentioning

confidence: 99%

Section: : Cyclic Lowest Density Mds Codes Withmentioning

confidence: 99%

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Cyclic Lowest Density MDS Array Codes

Cassuto

Bruck

2009

IEEE Trans. Inform. Theory

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“…Let F c and F d be two such 1-factors of K 2k such that c&d and 2k&1 are relatively prime. Consider an l-subset of those edges in F c that do not contain 1 . The sum of the vertices in these edges will be congruent to 2lc (mod 2k&1), since an edge xy in F c , x{ 1 { y, satisfies x+ y#2c (mod 2k&1).…”

Section: Pairwise Compatible Hamilton Path Decompositions Of K 2kmentioning

confidence: 99%

Pairwise Compatible Hamilton Decompositions ofKn

Verrall

1997

Journal of Combinatorial Theory, Series A

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“…In that paper, he also constructed a family of k many mutually Tx -complementary topologies on a set of cardinality k . Anderson then turned to the finite case [5], [6] and obtained some nice estimates of the maximum size of a mutually complementary family of topologies on a set of cardinality n . In fact, his topologies in the finite case 'are' equivalence relations.…”

Section: History and Introductionmentioning

confidence: 99%

Mutually complementary families of 𝑇₁ topologies, equivalence relations and partial orders

Steprāns¹,

Watson²

1995

Proc. Amer. Math. Soc.

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Abstract. We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of Tx topologies, the semi-lattice of partial orders and the lattice of equivalence relations. We show that there is a family of k many mutually complementary partial orders (and thus Tq topologies) on k and, using this family, build another family of k many mutually Tx complementary topologies on k . We obtain k many mutually complementary equivalence relations on any infinite cardinal k and thus obtain the simplest proof of a 1971 theorem of Anderson. We show that the maximum size of a mutually Tx complementary family of topologies on a set of cardinality k may not be greater than k unless co < k < 2C . We show that it is consistent with and independent of the axioms of set theory that there be N2 many mutually Tx-complementary topologies on wx using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations.

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Finite topologies and Hamiltonian paths

Cited by 49 publications

References 1 publication

Cyclic Lowest Density MDS Array Codes

Cyclic Lowest Density MDS Array Codes

Pairwise Compatible Hamilton Decompositions ofKn

Mutually complementary families of 𝑇₁ topologies, equivalence relations and partial orders

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