$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

2009 ◽  
Vol 54 (2) ◽  
pp. 167-179 ◽  
Author(s):  
J. Borges ◽  
C. Fernández-Córdoba ◽  
J. Pujol ◽  
J. Rifà ◽  
M. Villanueva
Keyword(s):  
Linear Codes ◽  
Generator Matrices

scholarly journals On Group Codes Over Elementary Abelian Groups

2003 ◽  
Vol 8 (2) ◽  
pp. 145
Author(s):  
Adnan A. Zain
Keyword(s):  
Abelian Group ◽  
Finite Fields ◽  
Linear Codes ◽  
Abelian Groups ◽  
Parity Check ◽  
Group Codes ◽  
New Codes ◽  
Generator Matrices

For group codes over elementary Abelian groups we present definitions of the generator and the parity check matrices, which are matrices over the ring of endomorphism of the group. We also lift the theorem that relates the parity check and the generator matrices of linear codes over finite fields to group codes over elementary Abelian groups. Some new codes that are MDS, self-dual, and cyclic over the Abelian group with four elements are given.

scholarly journals Counting Z2 Z4 Z8-additive Codes

2019 ◽  
Vol 12 (2) ◽  
pp. 668-679
Author(s):  
Basri Çalışkan ◽  
Kemal Balıkçı
Keyword(s):  
Linear Code ◽  
Coding Theory ◽  
Linear Codes ◽  
Algebraic Coding Theory ◽  
Additive Codes ◽  
Generator Matrices

In Algebraic Coding Theory, all linear codes are described by generator matrices. Any linear code has many generator matrices which are equivalent. It is important to find the number of the generator matrices for constructing of these codes. In this paper, we study Z_2 Z_4 Z_8-additive codes, which are the extension of recently introduced Z_2 Z_4-additive codes. We count the number of arbitrary Z_2 Z_4 Z_8-additive codes. Then we investigate connections to Z_2 Z_4 and Z_2 Z_8-additive codes with Z_2 Z_4 Z_8, and give some illustrative examples.

Linear codes over 𝔽4R and their MacWilliams identity

2020 ◽  
Vol 12 (06) ◽  
pp. 2050085
Author(s):  
Nasreddine Benbelkacem ◽  
Martianus Frederic Ezerman ◽  
Taher Abualrub
Keyword(s):  
Commutative Ring ◽  
Linear Codes ◽  
Inner Product ◽  
Dual Codes ◽  
Macwilliams Identity ◽  
Generator Matrices

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].

Two-Weight Linear Codes and Their Applications in Secret Sharing

2019 ◽  
Vol 28 (4) ◽  
pp. 706-711
Author(s):  
Yaru Wang ◽  
Fulin Li ◽  
Shixin Zhu
Keyword(s):  
Secret Sharing ◽  
Linear Codes

scholarly journals Multipreconditioned GMRES for simulating stochastic automata networks

2018 ◽  
Vol 16 (1) ◽  
pp. 986-998
Author(s):  
Chun Wen ◽  
Ting-Zhu Huang ◽  
Xian-Ming Gu ◽  
Zhao-Li Shen ◽  
Hong-Fan Zhang ◽  
...  
Keyword(s):  
Steady State ◽  
Probability Distribution ◽  
Linear Systems ◽  
Iterative Methods ◽  
Communication Systems ◽  
Queueing Systems ◽  
Steady State Probability ◽  
Stochastic Automata ◽  
Automata Networks ◽  
Generator Matrices

AbstractStochastic Automata Networks (SANs) have a large amount of applications in modelling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seidel are inefficient due to the huge size of the generator matrices. In this paper, the multipreconditioned GMRES (MPGMRES) is considered by using two or more preconditioners simultaneously. Meanwhile, a selective version of the MPGMRES is presented to overcome the rapid increase of the storage requirements and make it practical. Numerical results on two models of SANs are reported to illustrate the effectiveness of these proposed methods.

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