Recently Haah introduced a new quantum error correcting code embedded on a
cubic lattice. One of the defining properties of this code is the absence of
string logical operator. We present new codes with similar properties by
relaxing the condition on the local particle dimension. The resulting code is
well-defined when the local Hilbert space dimension is prime. These codes can
be divided into two different classes: the local stabilizer generators are
either symmetric or antisymmetric with respect to the inversion operation.
These is a nontrivial correspondence between these two classes. For any
symmetric code without string logical operator, there exists a complementary
antisymmetric code with the same property and vice versa. We derive a
sufficient condition for the absence of string logical operator in terms of the
algebraic constraints on the defining parameters of the code. Minimal number of
local particle dimension which satisfies the condition is 5. These codes have
logarithmic energy barrier for any logical error.Comment: 9 pages, 7 figure
