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- W2188591384 abstract "Upperboundsarederived forcodesintheGrass- mannmanifold withgiven minimumchordal distance. Theystem fromupperboundsforcodesintheproduct ofunitspheres and projective spaces. Thenewboundsareasymptotically better than thepreviously knownones. I.INTRODUCTION Theaiminthis paperistoobtain newupperboundsfor thesize ofcodesintheGrassmann manifold. Similar results canbeobtained also forcodesintheStiefel manifold. Codes intheGrassmann andStiefel manifolds haveapplications for MIMO systems (see (2)andreferences therein). Theapproach developed inthis paperistorelate theGrassmannian space toproducts ofunitspheres andprojective spaces. Thelinear programming method(call also Delsarte polynomial method) isthenapplied forproducts ofprojective spaces, yielding upperboundsonthesizeofcodesinthesespaces. This implies upperboundsonthesizeofGrassmannian codes, whoseasymptotic versions provide thebestcurrently known asymptotic bounds. Thepaperisorganized asfollows. Definitions andknown results aregiven inSection II. Section IIIdescribes relations between theGrassmann space andother spaces. Thesimplest oneconnects Grassmannian spaces totheunitsphere ofan asymptotically equal dimension; therefrom aboundthatal- ready improves upontheprevious ones(Theorem 3.2). Section IVdevelops theDelsarte polynomial methodforproducts of projective spaces, including theclassical methodthat involves theChristoffel-Darboux formula, andderives upper bounds for thesizeoftheassociated codes. A boundfortheasymptotic rateofthese codes isobtained. Section V discusses theconse- quences forGrassmannian codes. Inparticular, we showthat theboundobtained ontheasymptotic rateofGrassmannian codes fromtheproduct ofprojective spaces issometimes better thantheoneobtained inSection III. Duetospaceconsiderations, we present theresults only forreal Grassmannian codes. Theresults inthis paper canbe easily extended tothecomplex case, aswellastheStiefel manifold. Results regarding products ofprojective spaces can begeneralized toproducts ofallthecompact Riemannian manifolds thataretwo-point homogenous (these five spaces include theunit sphere andthecomplex projective space). See" @default.
- W2188591384 created "2016-06-24" @default.
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- W2188591384 date "2006-01-01" @default.
- W2188591384 modified "2023-09-27" @default.
- W2188591384 title "Bounds forCodes intheGrassmann Manifold" @default.
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