scholarly journals New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 101 ◽  
Author(s):  
Aixian Zhang ◽  
Zhe Ji
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Dual Codes ◽  
Self Duality ◽  
Reed Solomon Codes ◽  
Interesting Topic ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
New Construction

Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length n , so the problem of constructing q-ary MDS self-dual codes with various lengths is a very interesting topic. Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes.

Maximum Distance Separable Hankel Rhotrices over Finite Fields

2018 ◽  
Vol 43 (1-4) ◽  
pp. 13-45
Author(s):  
Prof. P. L. Sharma ◽  
◽  
Mr. Arun Kumar ◽  
Mrs. Shalini Gupta ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Maximum Distance Separable

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

scholarly journals On subsystem codes beating the quantum Hamming or Singleton bound

2007 ◽  
Vol 463 (2087) ◽  
pp. 2887-2905 ◽  
Author(s):  
Andreas Klappenecker ◽  
Pradeep Kiran Sarvepalli
Keyword(s):  
The Other ◽  
Maximum Distance ◽  
Open Problems ◽  
Prime Field ◽  
Stabilizer Codes ◽  
Singleton Bound ◽  
Other Hand ◽  
Maximum Distance Separable ◽  
Linear Subsystem ◽  
Stabilizer Code

Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

New LCD MDS codes constructed from generalized Reed–Solomon codes

2019 ◽  
Vol 18 (08) ◽  
pp. 1950150 ◽  
Author(s):  
Xueying Shi ◽  
Qin Yue ◽  
Shudi Yang
Keyword(s):  
Finite Field ◽  
Coding Theory ◽  
Theory And Practice ◽  
Maximum Distance ◽  
Mds Codes ◽  
Reed Solomon Codes ◽  
Maximum Distance Separable ◽  
Maximum Distance Separable Codes ◽  
Invertible Matrices

Maximum distance separable codes with complementary duals (LCD MDS codes) are very important in coding theory and practice, and have attracted a lot of attention. In this paper, we focus on LCD MDS codes constructed from generalized Reed–Solomon (GRS) codes over a finite field with odd characteristic. We detail two constructions of new LCD MDS codes, using invertible matrices and the roots of three classes of polynomials, respectively.

scholarly journals MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings

Information ◽  
2019 ◽  
Vol 10 (4) ◽  
pp. 153
Author(s):  
Sunghyu Han
Keyword(s):  
Maximum Distance ◽  
Galois Rings ◽  
Dual Codes ◽  
Maximum Distance Separable ◽  
Square Matrix

In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings G R ( p m , r ) with p ≡ − 1 ( mod 4 ) and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over G R ( p m , 3 ) with ( p = 3 and m = 2 , 3 , 4 , 5 , 6 ), ( p = 7 and m = 2 , 3 ), ( p = 11 and m = 2 ), ( p = 19 and m = 2 ), ( p = 23 and m = 2 ), and ( p = 31 and m = 2 ). In the building-up construction, it is important to determine the existence of a square matrix U such that U U T = − I , which is called an antiorthogonal matrix. We prove that there is no 2 × 2 antiorthogonal matrix over G R ( 2 m , r ) with m ≥ 2 and odd r.

scholarly journals PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION

2013 ◽  
Vol 11 (03) ◽  
pp. 1350027 ◽  
Author(s):  
MARTIANUS FREDERIC EZERMAN ◽  
SOMPHONG JITMAN ◽  
HAN MAO KIAH ◽  
SAN LING
Keyword(s):  
Error Correcting Codes ◽  
Complete Characterization ◽  
Maximum Distance ◽  
Mds Codes ◽  
Singleton Bound ◽  
Asymmetric Quantum ◽  
Maximum Distance Separable ◽  
Mds Conjecture ◽  
Css Construction ◽  
Quantum Error

Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

Optimal b-symbol constacyclic codes with respect to the Singleton bound

2019 ◽  
Vol 19 (08) ◽  
pp. 2050151 ◽  
Author(s):  
Hai Q. Dinh ◽  
Xiaoqiang Wang ◽  
Jirakom Sirisrisakulchai
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Maximum Distance ◽  
Constacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable

Let [Formula: see text] be the finite field of order [Formula: see text], where [Formula: see text] is a power of odd prime [Formula: see text]. Assume that [Formula: see text], [Formula: see text] are nonzero elements of the finite field [Formula: see text] such that [Formula: see text]. In this paper, we determine the [Formula: see text]-distance of [Formula: see text]-constacyclic codes with generator polynomials [Formula: see text] of length [Formula: see text], where [Formula: see text] and [Formula: see text]. As an application, all maximum distance separable (MDS) [Formula: see text]-symbol constacyclic codes of length [Formula: see text] over [Formula: see text] are established. Among other results, we construct several classes of new MDS symbol-pair codes with minimum symbol-pair distance six or seven by using repeated-root cyclic codes of length [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is an odd prime.

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