Some new entanglement-assisted quantum error-correcting MDS codes from generalized Reed–Solomon codes

Abstract

Subfield subcodes of Reed-Solomon codes and their duals, BCH codes, have been widely used for constructing quantum error-correcting codes with good parameters. In this paper, we study subfield subcodes of projective Reed-Solomon codes and their duals, we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature.

Entanglement-Assisted and Subsystem Quantum Codes: New Propagation Rules and Constructions

“…By determining the dimensions of the Hermitian or Euclidean hulls of MDS codes, many researchers have proposed various constructions of optimal EAQECCs. The works in [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23] are representatives of the literature on the topic.…”

“…Relying on (5), we turn the task of finding a nonzero codeword of E(D k,k−1 ) \ E(D k,k ) into counting the number of the roots of a polynomial. As the θ i,j in (18) traverses the elements of F q 2 , we obtain the following constructions.…”

“…Proof. By setting θ k−1,k−1 = 1 and θ t,t = θ i,j = 0, for each t ∈ [k, q − 1] and each (i, j) ∈ T in (18), there exists α −0(k−1)(q+1) , . .…”

“…Proof. In (18), let θ k−1,k−1 = 1, θ q−1,q−1 = −1, and θ t,t = θ i,j = 0 for each t ∈ [k, q − 2] and each (i, j) ∈ T . Then the corresponding codeword in…”

Two families of Entanglement-assisted Quantum MDS Codes from cyclic Codes

“…If C= [n, k, d] q 2 is a classical code over F q 2 and H is its parity check matrix, then C ⊥ h EA stabilizes an [[n, 2k − n + c, d; c]] q EAQECC, where c =rank(HH † ) is the number of maximally entangled states required and H † is the conjugate matrix of H over F q 2 . In resent years, scholars have constructed several entanglement-assisted quantum codes with good parameters in [1,32,11,16,18,17,6,10,33,26,27,4,34,22,7,28,31,25,36,8]. Many classes of EAQMDS codes have been constructed by different methods, in particular, by the Hermitian constructions from cyclic codes, constacyclic codes or negacyclic codes [4,5,34,21,20].…”