Triple Cyclic Codes Over 𝔽q + u𝔽q

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.

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

<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>

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

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.

Gray images of (1 − u)-constacyclic codes over the ring 𝔽pk + u𝔽pk + v𝔽pk + uv𝔽pk

Let [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a prime and [Formula: see text] is a positive integer. We define a gray map from a linear code of length [Formula: see text] over the ring [Formula: see text] to a linear code of length [Formula: see text] over the field [Formula: see text]. In this paper, we characterize the gray images of [Formula: see text]-constacyclic codes of an arbitrary length over the ring [Formula: see text] in terms of quasicyclic codes over [Formula: see text]. We obtain some optimal linear codes over [Formula: see text] as gray images.