Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
New code parameters from Reed-Solomon subfield codes
IEEE Transactions on Information Theory
Coset codes. I. Introduction and geometrical classification
IEEE Transactions on Information Theory - Part 1
Extending and LengtheningBCHCodes
Finite Fields and Their Applications
New Codes via the Lengthening of BCH Codes with UEP Codes
Finite Fields and Their Applications
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The idea behind the coset code construction(see [G.D. Forney, Coset Codes, IEEE Transactions onInformation Theory, Part I: Introduction and GeometricalClassification, pp. 1123–1151; Part II: Binary lattices andrelated codes, pp. 1152–1187; F.R. Kschischang and S. Pasupathy, IEEE Transactions on Information Theory 38 (1992), 227–246.]) is to reduce the construction of sphere packings toerror-correcting codes in a unified way. We give here a shortself-contained description of this method. In recent papers[J. Bierbrauer and Y. Edel, IEEE Transactions on InformationTheory 43 (1997), 953–968; J. Bierbrauer and Y. Edel, Finite Fields and Their Applications 3 (1997), 314–333; J. Bierbrauer and Y. Edel, IEEE Transactions on Information Theory 44 (1998), 1993; J. Bierbrauer, Y. Edel, and L. Tolhuizen, Finite Fields and Their Applications, submittedfor publication.] we constructed a large number of new binary,ternary and quaternary linear error-correcting codes. In a number ofdimensions our new codes yield improvements. Recently Vardy [A.Vardy, Inventiones Mathematicae 121, 119–134; A. Vardy, Density doubling, double-circulants, and new sphere packings,Trans. Amer. Math. Soc. 351 (1999), 271–283.] hasfound a construction, which yields record densitiesin dimensions 20, 27, 28, 29 and 30. We give a short description ofhis method using the language of coset codes. Moreover we are able toapply this method in dimension 18 as well, producing a sphere packingwith a record center density of (3/4)^9.