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In this note, the following basic question is explored: in a cyclic group,how are the Shannon entropies of the sum and difference of i.i.d. randomvariables related to each other? For the integer group, we show that they candiffer by any real number additively, but not too much multiplicatively; on theother hand, for $\mathbb{Z}/3\mathbb{Z}$, the entropy of the difference isalways at least as large as that of the sum. These results are closely relatedto the study of more-sum-than-difference (i.e. MSTD) sets in additivecombinatorics. We also investigate polar codes for $q$-ary input channels usingnon-canonical kernels to construct the generator matrix, and presentapplications of our results to constructing polar codes with significantlyimproved error probability compared to the canonical construction.

Entropies of weighted sums in cyclic groups and an application to polar codes