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The use of error-correcting codes for tight control of the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing (OFDM) transmission is considered. A well-known approach for the construction of such codes is to take a code that is good in the classical coding-theoretic sense and to choose a translate of this code that minimizes the PMEPR. A fundamental problem is to determine the minimum PMEPR over all translates of a given code. Motivated by a recent lower bound for this minimum, an existence result is presented here. Roughly speaking, given a code C of sufficiently large length n, there exists a translate of C with PMEPR at most k log(|C|n1+ϵ) for all ϵ > 0 and for some k independent of n. This result is then used to show that for n ≥ 32 there is a translate of the lengthened dual of a binary primitive t-error-correcting BCH code with PMEPR at most 8(t + 2)log n.