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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. random variables related to each other? For the integer group, we show that they can differ by any real number additively, but not too much multiplicatively; on the other hand, for Z/3Z, the entropy of the difference is always at least as large as that of the sum. These results are closely related to the study of more-sums-than-differences (i.e., MSTD) sets in additive combinatorics. We also investigate polar codes for q-ary input channels using non-canonical kernels to construct the generator matrix and present applications of our results to constructing polar codes with significantly improved error probability compared to the canonical construction. [ABSTRACT FROM AUTHOR]

This paper investigates entropic matroids, that is, matroids whose rank function is given as the Shannon entropy of random variables. In particular, we consider p-entropic matroids, for which the random variables each have support of cardinality p. We draw connections between such entropic matroids and secret-sharing matroids and show that entropic matroids are linear matroids when p = 2 , 3 but not when p = 9 . Our results leave open the possibility for p-entropic matroids to be linear whenever p is prime, with particular cases proved here. Applications of entropic matroids to coding theory and cryptography are also discussed. [ABSTRACT FROM AUTHOR]