A pure quantum state of N subsystems, each with d levels, is said to be k-uniform if all of its reductions to k qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of N qubits and 2-uniform states of N qutrits, and whether 3-uniform states of qudits (d > 2) for high values of N can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of N ≥ 8 qubits and 2uniform states of N qutrits for every N ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of N ≥ 8 qutrits, N = 6 and N ≥ 8 ququarts and ququints, N ≥ 6 qudits having d levels for any prime power d > 6, and N = 8 and N ≥ 12 qudits having d levels for non-prime-power d ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having d ≥ 4 levels. The proofs of existence of the 2-uniform states of N ≥ 6 qubits are simplified by using a class of OAs. Two special 3uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.
On the number of rotation symmetric Boolean functions SCIENCE CHINA Information Sciences 53, 537 (2010); Construction of rotation symmetric bent functions with maximumalgebraic degree SCIENCE CHINA Information Sciences 61, 038101 (2018);. LETTER. SCIENCE CHINA Information Sciences
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.