Traveling salesman problem in the space of binary vectors

The traveling salesman problem in the space Z₂ⁿ of all n-dimensional binary vectors can be formulated as follows. Given: a set C(n,M) of M n-dimensional vectors over GF(2) (M points in Z ₂ⁿ). Find: a Hamiltonian circuit for C(n,M) of minimum Hamming length, i.e., a cycle with minimum sum of Hamming distances between connected points (henceforth called minimal cycle). This problem is encountered in various situations related to the design and testing of computer hardware. In contrast with the “unrestricted” traveling salesman problem, the length of the minimal cycle in Z₂ⁿ is always upperbounded by that for the “worst” possible configuration of M points. Hence the authors come to the following variational problem. Problem: for any number M of points in Z₂ⁿ find the function L(n,M) defined as follows: M L(n,M)=max{C} min{P}Σ_i=1^M d(x_p(i),x_P(i+l(modM))) where x₁, ..., x _M are the points comprising a binary code C(n,M), d(x,y) is the Hamming distance between points x and y, {P} is the set of all possible permutations of the codewords, and the maximum is taken over the set of all possible binary codes of size M and dimension n. The problem is still open, but partial results are presented