Construction Of Perfect Codes Research Articles

On the construction of perfect codes from HEX signal constellations

Recently, minimum and non-minimum delay perfect codes were proposed for any channel of dimension n. Their construction appears in the literature as a subset of cyclic division algebras over Q(ζ3) only for the dimension n=2sn1, where s∈{0,1}, n1 is odd and the signal constellations are isomorphic to Z[ζ3]n. In this work, we propose an innovative methodology to extend the construction of minimum and non-minimum delay perfect codes as a subset of cyclic division algebras over Q(ζ3), where the signal constellations are isomorphic to the hexagonal A2n-rotated lattice, for any channel of any dimension n such that gcd(n,3)=1.

Perfect binary codes: constructions, properties, and enumeration

Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound the authors obtain a simple closed-form expression for the weight distribution of a perfect code. Furthermore, they prove that the characters of a perfect code satisfy certain constraints, and provide a sufficient condition for a binary code to be perfect. The latter result is employed to derive a generalization of the construction of Phelps (1983), which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions. Moreover, for any m/spl ges/4 the authors construct full-rank perfect binary codes of length 2/sup m/-1. These codes are obviously nonequivalent to any of the previously known perfect codes. Furthermore the latter construction exhibits the existence of full-rank perfect tilings. Finally, they construct a set of 2(2/sup cn/) nonequivalent perfect codes of length n, for sufficiently large n and a constant c=0.5-/spl epsiv/. Precise enumeration of the number of codes in this set provides a slight improvement over the results reported by Phelps. >

A Generalized Parity Function and Its Use in the Construction of Perfect Codes

Previous article Next article A Generalized Parity Function and Its Use in the Construction of Perfect CodesM. MollardM. Mollardhttps://doi.org/10.1137/0607013PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe define a new generalized parity function, and use it to obtain a product construction of single-error-correcting codes (binary or not).[1] R. W. Hamming, Error detecting and error correcting codes, Bell System Tech. J., 29 (1950), 147–160 12,35c CrossrefISIGoogle Scholar[2] H. S. Shapiro and , D. L. Slotnick, On the mathematical theory of error-correcting codes, IBM J. Res. Develop., 3 (1959), 25–34 20:5092 0109.10706 CrossrefISIGoogle Scholar[3] L. Vasilev, Jr., On ungrouped, close-packed codes, Problemy Kibernet. No., 8 (1962), 337–339 29:5661 Google Scholar[4] J. Schönheim, On linear and nonlinear single-error-correcting q-nary perfect codes, Information and Control, 12 (1968), 23–26 38:5517 0162.51203 CrossrefGoogle Scholar[5] K. T. Phelps, A general product construction for error correcting codes, SIAM J. Algebraic Discrete Methods, 5 (1984), 224–228 86c:94022 0546.94015 LinkISIGoogle Scholar[6] K. T. Phelps, A combinatorial construction of perfect codes, SIAM J. Algebraic Discrete Methods, 4 (1983), 398–403 85e:94024 0549.94029 LinkISIGoogle Scholar[7] Kevin T. Phelps, A product construction for perfect codes over arbitrary alphabets, IEEE Trans. Inform. Theory, 30 (1984), 769–771 10.1109/TIT.1984.1056963 86d:94028 0557.94012 CrossrefISIGoogle Scholar[8] M. Mollard, Une généralisation de la fonction parité, application á la construction de codes parfaits, R.R., 395, Laboratorire de Mathématiques Appliquées de Grenoble, Grenoble, France Google ScholarKeywordscombinatoricserror-correcting codes Previous article Next article FiguresRelatedReferencesCited byDetails On Perfect Codes and Tilings: Problems and Solutions1 August 2006 | SIAM Journal on Discrete Mathematics, Vol. 11, No. 2AbstractPDF (412 KB)Nonequivalent q-Ary Perfect Codes12 July 2006 | SIAM Journal on Discrete Mathematics, Vol. 9, No. 3AbstractPDF (874 KB)Tilings of Binary Spaces12 July 2006 | SIAM Journal on Discrete Mathematics, Vol. 9, No. 3AbstractPDF (2536 KB) Volume 7, Issue 1| 1986SIAM Journal on Algebraic Discrete Methods History Submitted:15 December 1983Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordscombinatoricserror-correcting codesPDF Download Article & Publication DataArticle DOI:10.1137/0607013Article page range:pp. 113-115ISSN (print):0196-5212ISSN (online):2168-345XPublisher:Society for Industrial and Applied Mathematics