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.

scholarly journals Nonexistence of some ternary linear codes with minimum weight -2 modulo 9

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Toshiharu Sawashima ◽  
Tatsuya Maruta
Keyword(s):  
Linear Code ◽  
Coding Theory ◽  
Linear Codes ◽  
Minimum Weight ◽  
Extension Theorem ◽  
Minimum Length ◽  
Geometric Methods ◽  
Optimal Linear ◽  
Ternary Linear Codes ◽  
Optimal Linear Codes

<p style='text-indent:20px;'>One of the fundamental problems in coding theory is to find <inline-formula><tex-math id="M3">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula>, the minimum length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> for which a linear code of length <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>, dimension <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>, and the minimum weight <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> over the field of order <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> exists. The problem of determining the values of <inline-formula><tex-math id="M9">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula> is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine <inline-formula><tex-math id="M10">\begin{document}$ n_3(6,d) $\end{document}</tex-math></inline-formula> for some values of <inline-formula><tex-math id="M11">\begin{document}$ d $\end{document}</tex-math></inline-formula> by proving the nonexistence of linear codes with certain parameters.</p>

scholarly journals Codes on Key Errors

2014 ◽  
Vol 14 (2) ◽  
pp. 31-37 ◽  
Author(s):  
P. K. Das
Keyword(s):  
Coding Theory ◽  
Linear Codes ◽  
Simultaneous Detection ◽  
Communication Channels ◽  
Upper Bounds ◽  
Parity Check ◽  
Lower And Upper Bounds ◽  
Different Types

Abstract Coding theory has started with the intention of detection and correction of errors which have occurred during communication. Different types of errors are produced by different types of communication channels and accordingly codes are developed to deal with them. In 2013 Sharma and Gaur introduced a new kind of an error which will be termed “key error”. This paper obtains the lower and upper bounds on the number of parity-check digits required for linear codes capable for detecting such errors. Illustration of such a code is provided. Codes capable of simultaneous detection and correction of such errors have also been considered.

scholarly journals Codes and Projective Multisets

10.37236/1375 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stefan Dodunekov ◽  
Juriaan Simonis
Keyword(s):  
Coding Theory ◽  
Linear Codes ◽  
Full Length ◽  
Self Duality ◽  
Projective Codes ◽  
Matrix Free

The paper gives a matrix-free presentation of the correspondence between full-length linear codes and projective multisets. It generalizes the Brouwer-Van Eupen construction that transforms projective codes into two-weight codes. Short proofs of known theorems are obtained. A new notion of self-duality in coding theory is explored.

scholarly journals Nonexistence Proofs for Five Ternary Linear Codes

2002 ◽  
Vol 1 (1) ◽  
pp. 35
Author(s):  
S. GURITMAN
Keyword(s):  
Linear Code ◽  
Minimum Distance ◽  
Linear Codes ◽  
Ternary Linear Codes

<p>An [n,k, dh-code is a ternary linear code with length n, dimension k and minimum distance d. We prove that codes with parameters [110,6, 72h, [109,6,71h, [237,6,157b, [69,7,43h, and [120,9,75h do not exist.</p>

Clique-Based Neural Associative Memories with Local Coding and Precoding

2016 ◽  
Vol 28 (8) ◽  
pp. 1553-1573 ◽  
Author(s):  
Asieh Abolpour Mofrad ◽  
Matthew G. Parker ◽  
Zahra Ferdosi ◽  
Mohammad H. Tadayon
Keyword(s):  
Neural Network ◽  
Associative Memory ◽  
Coding Theory ◽  
Simple Graph ◽  
Graph Representation ◽  
Associative Memories ◽  
Additive Codes ◽  
Partial Error ◽  
Pattern Retrieval ◽  
Local Coding

Techniques from coding theory are able to improve the efficiency of neuroinspired and neural associative memories by forcing some construction and constraints on the network. In this letter, the approach is to embed coding techniques into neural associative memory in order to increase their performance in the presence of partial erasures. The motivation comes from recent work by Gripon, Berrou, and coauthors, which revisited Willshaw networks and presented a neural network with interacting neurons that partitioned into clusters. The model introduced stores patterns as small-size cliques that can be retrieved in spite of partial error. We focus on improving the success of retrieval by applying two techniques: doing a local coding in each cluster and then applying a precoding step. We use a slightly different decoding scheme, which is appropriate for partial erasures and converges faster. Although the ideas of local coding and precoding are not new, the way we apply them is different. Simulations show an increase in the pattern retrieval capacity for both techniques. Moreover, we use self-dual additive codes over field [Formula: see text], which have very interesting properties and a simple-graph representation.

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