Closure With A Twist - Cwatset

In mathematics, a cwatset is a set of bitstrings, all of the same length, which is closed with a twist.

If each string in a cwatset, C, say, is of length n, then C will be a subset of Z₂n. Thus, two strings in C are added by adding the bits in the strings modulo 2 (that is, addition without carry, or exclusive disjunction). The symmetric group on n letters, Sym(n), acts on Z₂n by bit permutation:

p((c₁,...,c_n))=(c_p(1),...,c_p(n)),

where c=(c₁,...,c_n) is an element of Z₂n and p is an element of Sym(n). Closure with a twist now means that for each element c in C, there exists some permutation p_c such that, when you add c to an arbitrary element e in the cwatset and then apply the permutation, the result will also be an element of C. That is, denoting addition without carry by +, C will be a cwatset if and only if

This condition can also be written as