An Improved Upper Bound of the Rate of Euclidean Superimposed Codes

A family of n-dimensional unit norm vectors is a Euclidean superimposed code, if the sums of any two distinct at most m-tuples of vectors are separated by a certain minimum Euclidean distance d. Ericson and Gyorfi (see IEEE Trans. Inform. Theory, vol.34, no.4, p.877-80, 1988) proved that the rate of such a code is between (log m)/4m and (logm)/m for m large enough. In this paper-improving the above long-standing best upper bound for the rate-it is shown that the rate is always at most (log m)/2m, i.e., the size of a possible superimposed code is at most the root of the size given by Ericson and Gyorfi. We also generalize these codes to other normed vector spaces.

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