Quantum Error-Correcting Codes Based on Orthogonal Arrays

In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N, K, d + 1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N, 1, d + 1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3, s, 2))s for all s(i) = 3, ((4, s(2), 2))s for all s(i) = 5, ((5, s, 3))s for all s(i) = 4, ((6, s(2), 3))s for all s(i) = 5, ((7, s(3), 3))s for all s(i) = 7, ((8, s(2), 4))s for all s(i) = 9, ((9, s(3), 4))s for all s(i)= 11, ((9, s, 5))s for all s(i) = 9, ((10, s(2), 5))s for all s(i )= 11, ((11, s, 6))s for all s(i) = 11, and ((12, s(2), 6))s for all s(i) = 13, where s = s(1) . . . s(n) and s(1), ... , s(n) are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.

Automated summary

In this paper, a relation between quantum error-correcting codes and orthogonal arrays with orthogonal partitions is established using the Hamming distance. This relation is a generalization of the existing relation between quantum error-correcting codes and irredundant orthogonal arrays. The construction of pure quantum error-correcting codes is made possible through this relation, leading to the explicit construction of numerous optimal quantum codes. The advantages of this approach include the constructive nature of the results, the purity of the constructed codes, and the reduced number of terms in each basis state. Furthermore, the method developed can be extended to the construction of quantum error-correcting codes over mixed alphabets.