79 research outputs found
Stabilizer quantum codes from -affine variety codes and a new Steane-like enlargement
New stabilizer codes with parameters better than the ones available in the
literature are provided in this work, in particular quantum codes with
parameters and that are records.
These codes are constructed with a new generalization of the Steane's
enlargement procedure and by considering orthogonal subfield-subcodes --with
respect to the Euclidean and Hermitian inner product-- of a new family of
linear codes, the -affine variety codes
Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance
Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance
d. We study the minimum asymptotic redundancy \rho(q,n,d)=n-log_q A(q,n,d) as n
grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are
designed that improve on the BCH codes and have the lowest asymptotic
redundancy \rho(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this
work, codes of fixed distance that asymptotically surpass BCH codes and the
Gilbert-Varshamov bound were designed only for distances 4,5 and 6.Comment: Submitted to IEEE Trans. on Info. Theor
New Quantum Codes from Evaluation and Matrix-Product Codes
Stabilizer codes obtained via CSS code construction and Steane's enlargement
of subfield-subcodes and matrix-product codes coming from generalized
Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes
with good quantum parameters are supplied, in particular, some binary codes of
lengths 127 and 128 improve the parameters of the codes in
http://www.codetables.de. Moreover, non-binary codes are presented either with
parameters better than or equal to the quantum codes obtained from BCH codes by
La Guardia or with lengths that can not be reached by them
Low-complexity quantum codes designed via codeword-stabilized framework
We consider design of the quantum stabilizer codes via a two-step,
low-complexity approach based on the framework of codeword-stabilized (CWS)
codes. In this framework, each quantum CWS code can be specified by a graph and
a binary code. For codes that can be obtained from a given graph, we give
several upper bounds on the distance of a generic (additive or non-additive)
CWS code, and the lower Gilbert-Varshamov bound for the existence of additive
CWS codes. We also consider additive cyclic CWS codes and show that these codes
correspond to a previously unexplored class of single-generator cyclic
stabilizer codes. We present several families of simple stabilizer codes with
relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl
Quantum codes from affine variety codes and their subfield-subcodes
We use affine variety codes and their subfield-subcodes to obtain quantum stabilizer codes via the CSS code construction. With this procedure we get codes with good parameters, some of them exceeding the CSS quantum Gilbert–Varshamov bound given by Feng and Ma
On Hull-Variation Problem of Equivalent Linear Codes
The intersection () of a linear code and its Euclidean dual (Hermitian dual ) is called the Euclidean
(Hermitian) hull of this code. The construction of an entanglement-assisted
quantum code from a linear code over or depends
essentially on the Euclidean hull or the Hermitian hull of this code. Therefore
it is natural to consider the hull-variation problem when a linear code is transformed to an equivalent code . In this paper
we introduce the maximal hull dimension as an invariant of a linear code with
respect to the equivalent transformations. Then some basic properties of the
maximal hull dimension are studied. A general method to construct
hull-decreasing or hull-increasing equivalent linear codes is proposed. We
prove that for a nonnegative integer satisfying , a
linear self-dual code is equivalent to a linear -dimension hull
code. On the opposite direction we prove that a linear LCD code over satisfying and is equivalent to a linear
one-dimension hull code under a weak condition. Several new families of
negacyclic LCD codes and BCH LCD codes over are also constructed.
Our method can be applied to the generalized Reed-Solomon codes and the
generalized twisted Reed-Solomon codes to construct arbitrary dimension hull
MDS codes. Some new EAQEC codes including MDS and almost MDS
entanglement-assisted quantum codes are constructed. Many EAQEC codes over
small fields are constructed from optimal Hermitian self-dual codes.Comment: 33 pages, minor error correcte
Classical and Quantum Evaluation Codesat the Trace Roots
We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field F 4 which are records at http://codetables.de/
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