Dualities over the Cross Product of the Cyclic Groups of Order 2

We determine the number of symmetric dualities on the s-fold cross product of the cyclic group of order 2, which is the additive group of the finite field F2s. We show that the ratio of symmetric dualities over all dualities goes to 0 as s goes to infinity.We also prove a surprising result that given any two binary codes C and D of the same length n with |C||D| = 2n, then viewing them as groups there is a symmetric duality M with CM = D, which also relates their weight enumerators as additive codes in a group via the MacWilliams relations. Using this theorem we show that any additive code in this setting can be viewed as an additive complementary dual code of length 1 with respect to some duality.