Dualities over the cross product of the cyclic groups of order 2

Abstract

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 $ {\mathbb{F}}_{2^s}. $ 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| = 2^n $, then viewing them as groups there is a symmetric duality $ M $ with $ C^M = 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.