11 research outputs found
Some constructions of quantum MDS codes
We construct quantum MDS codes with parameters for all , . These codes are shown to exist by
proving that there are classical generalised Reed-Solomon codes which contain
their Hermitian dual. These constructions include many constructions which were
previously known but in some cases these codes are new. We go on to prove that
if then there is no generalised Reed-Solomon
code which contains its Hermitian dual. We also construct
an quantum MDS code, an quantum
MDS code and a quantum MDS code, which are the first
quantum MDS codes discovered for which , apart from the quantum MDS code derived from Glynn's code
Some constructions of quantum MDS codes
The version of record os available online at: http://dx.doi.org/10.1007/s10623-021-00846-yWe construct quantum MDS codes with parameters [[q2+1,q2+3-2d,d]]q for all d¿q+1, d¿q. These codes are shown to exist by proving that there are classical generalised Reed–Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if d¿q+2 then there is no generalised Reed–Solomon [n,n-d+1,d]q2 code which contains its Hermitian dual. We also construct an [[18,0,10]]5 quantum MDS code, an [[18,0,10]]7 quantum MDS code and a [[14,0,8]]5 quantum MDS code, which are the first quantum MDS codes discovered for which d¿q+3, apart from the [[10,0,6]]3 quantum MDS code derived from Glynn’s code.Postprint (author's final draft
New Quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes
Quantum maximum-distance-separable (MDS for short) codes are an important
class of quantum codes. In this paper, by using Hermitian self-orthogonal
generalized Reed-Solomon (GRS for short) codes, we construct four new classes
of -ary quantum MDS codes. The -ary quantum MDS codes we construct have
larger minimum distances. And the minimum distance of these codes is greater
than . Furthermore, it turns out that our quantum MDS codes generalize
the previous conclusions.Comment: 19 pages, 2 table
Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal
© 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksWe prove that there is a Hermitian self-orthogonal k -dimensional truncated generalised Reed-Solomon code of length n¿q2 over Fq2 if and only if there is a polynomial g¿Fq2 of degree at most (q-k)q-1 such that g+gq has q2-n distinct zeros. This allows us to determine the smallest n for which there is a Hermitian self-orthogonal k -dimensional truncated generalised Reed-Solomon code of length n over Fq2 , verifying a conjecture of Grassl and Rötteler. We also provide examples of Hermitian self-orthogonal k -dimensional generalised Reed-Solomon codes of length q2+1 over Fq2 , for k=q-1 and q an odd power of two.Peer ReviewedPostprint (author's final draft
Determining hulls of generalized Reed-Solomon codes from algebraic geometry codes
In this paper, we provide conditions that hulls of generalized Reed-Solomon
(GRS) codes are also GRS codes from algebraic geometry codes. If the conditions
are not satisfied, we provide a method of linear algebra to find the bases of
hulls of GRS codes and give formulas to compute their dimensions. Besides, we
explain that the conditions are too good to be improved by some examples.
Moreover, we show self-orthogonal and self-dual GRS codes