Optimal coding with one asymmetric error: Below the sphere packing bound

Abstract

Ulam and Rényi asked what is the minimum number of yesno questions needed to nd 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 nitely many exceptional m, one can nd x asking Berlekamp’s minimum number ql(m) of questions, i.e., the smallest integer q satisfying the sphere packing bound for error-correcting codes. The Ulam-Rényi problem amounts to nding optimal error-correcting codes for the binary symmetric channel with noiseless feedback, rst 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-Renyi problem, asking for the minimum number of yes-no questions needed to nd 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 innitely 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.