Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1

Abstract

he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.

A new method for constructing EAQEC MDS codes

“…In Table 3, we give our general conclusions to make comparisons with those known results in Refs. [6,9,18,22,25,[28][29][30][31]. The results show that the lengths and entanglement bits of those known conclusions above EAQEC MDS codes studied in the literatures are fixed.…”

“…However, the lengths of two classes of EAQEC MDS codes derived from our construction are very flexible, and the entanglement bits of three classes of EAQEC MDS codes derived from our construction are very flexible. Constraints Distance References [[ q−1 at , q−1 at − 2d + 6, d; 4]]q q = l 2 , l = atm + 1 is an odd prime power, a be even, or a be odd and t be even [[q + 1, q + 1 − 2d + 6, d; 4]]q q = p 2a , p a ≡ 1 (mod 4) p a + 3 ≤ d ≤ 3p a − 1 and d is even [6] [[q + 1, q + 1 − 2d + 3, d; 1]]q q = p 2a 2 ≤ d ≤ p a and d is even [9] [[q + 1, q − 2d + 11; d, 9]]q q = p 2a , p a ≡ 3 (mod 4), p a > 7 2p a + 4 ≤ d ≤ 4p a − 2 even [28] [[q + 1, q − d + 2, d; d − 1]]q q = p 2a , r | p a − 1 and r ∤ p a + 1 2 ≤ d ≤ (r−1)(p 2a −1) 2 + 2…”

“…Fujiwara et al [8] gave a general method for constructing entanglement-assisted quantum low-density parity check codes. Fan, Chen and Xu [9] provided a construction of entanglement-assisted quantum maximum distance separable (EAQEC MDS) codes with a small number c of pre-shared maximally entangled states. From constacyclic codes, Chen et al and Lu et al constructed new EAQEC MDS codes with larger minimum distance and consumed 4 entanglement bits in [6,23], respectively.…”

Entanglement-assisted Quantum MDS Codes from generalized Reed-Solomon Codes

“…In [5], a new decomposition of negacyclic codes is proposed, by which four new classes of EAQEC codes have been constructed. In [6], Fan et al constructed some classes of EAQMDS codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. In [14], Qian and Zhang constructed some new classes of MDS linear complementary dual(LCD) codes with respect to Hermitian inner product.…”

New Classes of Entanglement-assisted Quantum MDS Codes

“…In [5], new decomposition of negacyclic codes are proposed, by which four new classes of EAQEC codes have been constructed. In [6], Fan et al have constructed some classes of EAQMDS codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. In [19], Qian and Zhang have constructed some new classes of MDS liner complementary dual(LCD) codes with respect to Hermitian inner product.…”