Abstract
Perfect multiple coverings generalize the concept of perfect codes by allowing for multiplicities, much as t-designs generalize Steiner systems. A necessary and sufficient condition is found to determine when a metric scheme admits a nontrivial perfect multiple covering. Results specific to the classical Hamming and Johnson schemes are given which bear out the relationship between t-designs, orthogonal arrays, and perfect multiple coverings.
This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation
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References
E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, California, 1984.
N. L. BIGGS, Perfect Codes in graphs, J. Combinatorial Theory, Ser. B, 15 (1973), pp. 289–296.
N. L. BIGGS, Perfect codes and distance-transitive graphs, in Combinatorics (V. C. Mavron and T. P. McDonough, eds.), L. M. S. Lecture Note Ser. 13, 1974, pp. 1–8.
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance Regular Graphs, Springer-Verlag, New York, 1989.
L. Chihara, On the zeros of the Askey-Wilson polynomials, with applications to coding theory, SIAM J. Math. Anal., 18 (1987), pp. 191–207.
L. Chihara and D. Stanton, Zeros of generalized Krawtchouk polynomials, J. Approx. Th. (to appear).
R. Clayton, Multiple packings and coverings in algebraic coding theory, Ph.D. thesis, University of California, Los Angeles, 1987.
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements no. 10, (1973).
P. Diaconis and R. L. Graham, The Radon transform on Zk 2, Pac. J. Math., 118 (1985), pp. 323–345.
Y. Hong, On the nonexistence of nontrivial perfect e-codes and tight 2e-designs in Hamming schemes H(n,q) with e ≥ 3 and q ≥ 3, Graphs and Combinatorics, 2 (1986), pp. 145–164.
J. H. Van Lint, On the nonexistence of perfect 2- and 3-Hamming-error-correcting codes over GF(q), Info, and Control, 16 (1970), pp. 396–401.
J. H. Van Lint, Introduction to Coding Theory, Springer-Verlag, New York, 1982.
A. TietÄvÄinen, On the nonexistence of perfect codes over finite fields, SIAM J. Applied Math., 24 (1973), pp. 88–96.
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© 1990 Springer-Verlag New York, Inc.
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Clayton, R. (1990). Perfect Multiple Coverings in Metric Schemes. In: Coding Theory and Design Theory. The IMA Volumes in Mathematics and Its Applications, vol 20. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8994-1_5
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DOI: https://doi.org/10.1007/978-1-4613-8994-1_5
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