Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance

Abstract

The entanglement-assisted (EA) formalism allows arbitrary classical linear codes to transform into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we propose a decomposition of the defining set of constacyclic codes. Using this method, we construct four classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper limit greater than 3q − 1. Some of them have much larger minimum distance than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

Two Families of Entanglement-Assisted Quantum MDS Codes from Constacyclic Codes

“…However, a defining set T of a non-dual-containing (or non-self-orthogonal) classical codes is T ∩T −q = ∅. In order to construct EA-quantum MDS codes for larger distance than q + 1 of code length n ≤ q 2 + 1, we recall the fundamental definition of decomposition of the defining set of constacyclic codes [35]. There are also other types of definition for decomposition of the defining set of cyclic codes, negacyclic codes, see [14,26,33,34].…”

“…Lemma 2.4 [35]. Let T be a defining set of a constacyclic code C, T = T ss ∪ T sas be decomposition of T .…”

“…In resent years, lots of scholars have constructed many entanglement-assisted quantum codes with good parameters, see [4, 6-11, 13-16, 31-41] . In [35], we proposed the concept about a decomposition of the defining set of constacyclic codes. With the help of this concept, we construct some good entanglement-assisted quantum MDS codes.…”

Some New Classes of Entanglement-Assisted Quantum MDS Codes Derived From Constacyclic Codes

“…α , which is different from the one used in [11], [28]. Additionally, by the method, the length of entanglement-assisted quantum codes is more general, so we can obtain more entanglement-assisted quantum MDS codes with minimum distance that is more than q 2 + 1 relative to the ones of [19], [26], [27]. Furthermore, we can also use the same method of the decomposition of the defining set of constacyclic codes to obtain other entanglement-assisted quantum MDS codes with the number of pre-shared maximally entangled states that exceeds 9 in the Hermitian construction.…”

“…Based on the results of [22], [25], we proposed a decomposition of the defining set of negacyclic codes and utilized this method to construct some families of entanglement-assisted quantum MDS codes with different lengths in [6]. In [26], [27], Lü et al used the decomposition of the defining set of negacyclic codes and constacyclic codes to construct some families of entanglement-assisted quantum MDS codes respectively, and someone of those constructed quantum MDS codes have larger minimum distance with d ≥ q + 1. In [23], Liu et al constructed some new entanglement-assisted quantum MDS codes from constacyclic codes of length n = q 2 −1 r for r = 3, 5, 6, 7 and q ≡ −1 mod r. In fact, pre-shared entanglement can improve the error-correcting ability of quantum codes.…”