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Full rank perfect codes and alpha-kernels
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2009 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 8, 2202-2216 p.Article in journal (Refereed) Published
Abstract [en]

A perfect 1-error correcting binary code C, perfect code for short, of length n = 2(m) - 1 has full rank if the linear span < C > of the words of C has dimension n as a vector space over the finite field F-2. There are just a few general constructions of full rank perfect codes, that are not given by recursive methods using perfect codes of length shorter than n. In this study we construct full rank perfect codes, the so-called normal alpha-codes, by first finding the superdual of the perfect code. The superdual of a perfect code consists of two matrices G and T in which simplex codes play an important role as subspaces of the row spaces of the matrices G and T. The main idea in our construction is the use of alpha-words. These words have the property that they can be added to certain rows of generator matrices of simplex codes such that the result will be (other) sets of generator matrices for simplex codes. The kernel of these normal alpha-codes will also be considered. It will be proved that any subspace, of the kernel of a normal alpha-code, that satisfies a certain property will be the kernel of another perfect code, of the same length n. In this way, we will be able to relate some of the full rank perfect codes of length n to other full rank perfect codes of the same length n.

Place, publisher, year, edition, pages
2009. Vol. 309, no 8, 2202-2216 p.
Keyword [en]
Perfect codes, error-correcting codes, tilings, construction, spaces
Identifiers
URN: urn:nbn:se:kth:diva-18348DOI: 10.1016/j.disc.2008.04.051ISI: 000265176000028Scopus ID: 2-s2.0-62849115982OAI: oai:DiVA.org:kth-18348DiVA: diva2:336394
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2010-12-23Bibliographically approved

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