scholarly journals On Optimal Linear Codes over ${\mathbb{F}}_8$

10.37236/521 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Rie Kanazawa ◽  
Tatsuya Maruta
Keyword(s):  
Lower Bounds ◽  
Linear Codes ◽  
Upper And Lower Bounds ◽  
Extension Theorems ◽  
Geometric Methods ◽  
Optimal Linear ◽  
Optimal Linear Codes

Let $n_q(k,d)$ be the smallest integer $n$ for which there exists an $[n,k,d]_q$ code for given $q,k,d$. It is known that $n_8(4,d) = \sum_{i=0}^{3} \left\lceil d/8^i \right\rceil$ for all $d \ge 833$. As a continuation of Jones et al. [Electronic J. Combinatorics 13 (2006), #R43], we determine $n_8(4,d)$ for 117 values of $d$ with $113 \le d \le 832$ and give upper and lower bounds on $n_8(4,d)$ for other $d$ using geometric methods and some extension theorems for linear codes.

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 Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 15
Author(s):  
Lucky Galvez ◽  
Jon-Lark Kim
Keyword(s):  
Linear Codes ◽  
Cyclic Codes ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Decoding Algorithm ◽  
Decoding Algorithms ◽  
Syndrome Decoding ◽  
Optimal Linear ◽  
Optimal Linear Codes ◽  
Efficient Decoding

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.

scholarly journals On optimal linear codes over F5

2009 ◽  
Vol 309 (6) ◽  
pp. 1255-1272 ◽  
Author(s):  
Tatsuya Maruta ◽  
Maori Shinohara ◽  
Ayako Kikui
Keyword(s):  
Linear Codes ◽  
Optimal Linear ◽  
Optimal Linear Codes

Triple Cyclic Codes Over 𝔽q + u𝔽q

2021 ◽  
pp. 1-21
Author(s):  
Ting Yao ◽  
Shixin Zhu ◽  
Binbin Pang
Keyword(s):  
Special Class ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Cyclic Shift ◽  
Generating Sets ◽  
Optimal Linear ◽  
Number Formula ◽  
The Relationship ◽  
Optimal Linear Codes

Let [Formula: see text], where [Formula: see text] is a power of a prime number [Formula: see text] and [Formula: see text]. A triple cyclic code of length [Formula: see text] over [Formula: see text] is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as [Formula: see text]-submodules of [Formula: see text]. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over [Formula: see text] are discussed briefly in the end.

scholarly journals Weight Distribution of Periodic Errors and Optimal/Anti-Optimal Linear Codes

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Pankaj Kumar Das
Keyword(s):  
Weight Distribution ◽  
Linear Codes ◽  
Parity Check ◽  
Optimal Linear ◽  
Optimal Linear Codes

The paper discusses weight distribution of periodic errors and then the optimal case on bounds of parity check digits for (n=n1+n2,k) linear codes overGF(q)that corrects all periodic errors of orderrin the first block of lengthn1and all periodic errors of ordersin the second block of lengthn2and no others. Further, we extend the study to the case when the errors are in the form of periodic errors of orderr(ands) or more in the two subblocks.

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