다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

We addressthe question: How should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The resuIts of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension ½(m–l)(m+2), which provides a (usually) lowerdimensional representation than the Plucker embedding, and leads to a proof that many of the new packings are optimal The results have applications to the graphical display of multidimensional data via Asimov's grand tour method

/pdf/packing-lines-planes-etc-packings-in-grassmannian-spaces-59ara3fv0c.pdf

Packing lines, planes, etc.: packings in Grassmannian spaces

This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension (m-1)(m+2)/2, which provides a (usually) lower-dimensional representation than the Pluecker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's "Grand Tour" method.

Packing Lines, Planes, etc.: Packings in Grassmannian Space

In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.

A new family of linear maximum rank distance codes

Generalized Twisted Gabidulin Codes

A new family of MRD-codes

We introduce a family of linear sets of $\mathrm{PG}(1,q^{2n})$ arising from maximum scattered linear sets of pseudoregulus type of $\mathrm{PG}(3,q^{n})$. For $n=3,4$ and for certain values of the parameters we show that these linear sets of $\mathrm{PG}(1,q^{2n})$ are maximum scattered and they yield new MRD-codes with parameters $(6,6,q;5)$ for $q>2$ and with parameters $(8,8,q;7)$ for $q$ odd.

Let L be a linear set of pseudoregulus type in a line $$\ell $$l in $$\varSigma ^*={\mathrm {PG}}(t-1,q^t)$$Σź=PG(t-1,qt), $$t=5$$t=5 or $$t>6$$t>6. We provide examples of q-order canonical subgeometries $$\varSigma _1,\, \varSigma _2 \subset \varSigma ^*$$Σ1,Σ2źΣź such that there is a $$(t-3)$$(t-3)-subspace $$\varGamma \subset \varSigma ^*\setminus (\varSigma _1 \cup \varSigma _2 \cup \ell )$$ΓźΣź\(Σ1źΣ2źl) with the property that for $$i=1,2$$i=1,2, L is the projection of $$\varSigma _i$$Σi from center $$\varGamma $$Γ and there exists no collineation $$\phi $$ź of $$\varSigma ^*$$Σź such that $$\varGamma ^{\phi }=\varGamma $$Γź=Γ and $$\varSigma _1^{\phi }=\varSigma _2$$Σ1ź=Σ2. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des Codes Cryptogr 56:89---104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.

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On the equivalence of linear sets

We generalize the example of linear set presented by the last two authors in “Vertex properties of maximum scattered linear sets of PG (1,  q n ) " (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are new. This solves an open problem posed in “Vertex properties of maximum scattered linear sets of PG (1,  q n ) " (2019). As a consequence of Sheekey’s results in “A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters (6, 6,  q ; 5) .

/pdf/a-new-family-of-maximum-scattered-linear-sets-in-pg-1-q-6-3qnant6nel.pdf

Corrado Zanella

Papers

A new family of MRD-codes

A new family of MRD-codes

On the equivalence of linear sets

A new family of maximum scattered linear sets in PG(1, q 6 )

On scattered linear sets of pseudoregulus type in PG(1,qt)