The modulo-sum problem, proposed by Körner and Marton (KM), is a representative problem in the field of distributed source coding. In the modulo-sum problem, two correlated sources are encoded separately, and the decoder decodes the modulo-sum of the sources. It is clear that the Slepian-Wolf (SW) coding rate region is achievable for the modulo-sum problem. Körner and Marton proved that the SW coding rate region can be improved by structured coding based on linear codes. Ahlswede and Han (AH) proposed AH coding, which combines structured coding and random coding, and expressed its rate using auxiliary random variables. However, it was conjectured that the minimum sum rate of AH coding cannot be smaller than the minimum of the sum rates achievable by KM coding or SW coding. Subsequently, Kakishima and Watanabe considered a multi-letter extension of AH coding, and designed the auxiliary random variables by taking the XOR of adjacent bits of the source sequences. Through numerical computation, they demonstrated the existence of source parameters for which multi-letter AH coding improves upon SW coding. However, this confirmation remained numerical, and the conditions under which multi-letter extended AH coding improves upon SW coding have not been analytically characterized. In this study, we analytically characterize the conditions under which multi-letter extended AH coding improves upon SW coding. Our condition is tight in the sense that it coincides with the complement of the known SW optimal sufficient condition. To obtain the conditions, we apply the method of types to reduce the evaluation of the multi-letter expression to a comparison of single-letter divergences, which might be of independent interest.

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