Abstract
Ulam and Rényi asked what is the minimum number of yes-no questions needed to find an unknown m-bit number x, if up to l of the answers may be erroneous/mendacious. For each l it is known that, up to only finitely many exceptional m, one can find x asking Berlekamp’s minimum number q ℓ (m) of questions, i.e., the smallest integer q satisfying the sphere packing bound for error-correcting codes. The Ulam-Rényi problem amounts to finding optimal error-correcting codes for the binary symmetric channel with noiseless feedback, first considered by Berlekamp. In such concrete situations as optical transmission, error patterns are highly asymmetric—in that only one of the two bits can be distorted. Optimal error-correcting codes for these asymmetric channels with feedback are the solutions of the half-lie variant of the Ulam-Rényi problem, asking for the minimum number of yes-no questions needed to find an unknown m-bit number x, if up to l of the negative answers may be erroneous/mendacious. Focusing attention on the case l = 1; in this self-contained paper we shall give tight upper and lower bounds for the half-lie problem. For infinitely many m’s our bounds turn out to be matching, and the optimal solution is explicitly given, thus strengthening previous estimates by Rivest, Meyer et al.
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References
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Cicalese, F., Mundici, D. (2000). Optimal Coding with One Asymmetric Error: Below the Sphere Packing Bound. In: Du, DZ., Eades, P., Estivill-Castro, V., Lin, X., Sharma, A. (eds) Computing and Combinatorics. COCOON 2000. Lecture Notes in Computer Science, vol 1858. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44968-X_16
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DOI: https://doi.org/10.1007/3-540-44968-X_16
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