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CYCLIC AND CONSTACYCLIC SELF-DUAL CODES OVER R<sub>k</sub>
Karadeniz, Suat,Kelebek, Ismail Gokhan,Yildiz, Bahattin Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4
In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1={\mathbb{F}}_2+u{\mathbb{F}}_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in [1] and we explain why their claim fails.
Batra, Sudhir,Mathur, Rekha Korean Mathematical Society 2018 대한수학회보 Vol.55 No.3
The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime p are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^n$, $n{\geq}1$ over the quaternary field $F_4$ are obtained. These codes are observed to be subcodes of Q codes of length $p^n$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a Q code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic Q codes of order 6 exist.
Cyclic and constacyclic self-dual codes over R_k
Suat Karadeniz,Ismail Gokhan Kelebek,Bahattin Yildiz 대한수학회 2017 대한수학회보 Vol.54 No.4
In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1 = \F_2+u\F_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in \cite{Batoul} and we explain why their claim fails.
Primitive idempotents in the ring $F_{4}[x]/\langle x^{p^{n}}-1\rangle$ and cyclotomic $Q$ codes
Sudhir Batra,Rekha Mathur 대한수학회 2018 대한수학회보 Vol.55 No.3
The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime $p$ are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^{n}$, $n\ge 1$ over the quaternary field $F_{4}$ are obtained. These codes are observed to be subcodes of $Q$ codes of length $p^{n}$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a $Q$ code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic $Q$ codes of order 6 exist.
SKEW CYCLIC CODES OVER F<sub>p</sub> + vF<sub>p</sub>
Gao, Jian The Korean Society for Computational and Applied M 2013 Journal of applied mathematics & informatics Vol.31 No.3
In this paper, we study a special class of linear codes, called skew cyclic codes, over the ring $R=F_p+vF_p$, where $p$ is a prime number and $v^2=v$. We investigate the structural properties of skew polynomial ring $R[x,{\theta}]$ and the set $R[x,{\theta}]/(x^n-1)$. Our results show that these codes are equivalent to either cyclic codes or quasi-cyclic codes. Based on this fact, we give the enumeration of distinct skew cyclic codes over R.
Kewat, Pramod Kumar,Kushwaha, Sarika Korean Mathematical Society 2018 대한수학회보 Vol.55 No.1
Let $R_{u{^2},v^2,w^2,p}$ be a finite non chain ring ${\mathbb{F}}_p[u,v,w]{\langle}u^2,\;v^2,\;w^2,\;uv-vu,\;vw-wv,\;uw-wu{\rangle}$, where p is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u{^2},v^2,w^2,p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length 8n over ${\mathbb{F}}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.
QUADRATIC RESIDUE CODES OVER ℤ<sub>9</sub>
Taeri, Bijan Korean Mathematical Society 2009 대한수학회지 Vol.46 No.1
A subset of n tuples of elements of ${\mathbb{Z}}_9$ is said to be a code over ${\mathbb{Z}}_9$ if it is a ${\mathbb{Z}}_9$-module. In this paper we consider an special family of cyclic codes over ${\mathbb{Z}}_9$, namely quadratic residue codes. We define these codes in term of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over finite fields.
Cyclic codes over the ring ${\mathbb F}_p[u,v,w]/\langle u^2, v^2, w^2, uv-vu, vw-wv, uw-wu \rangle$
Pramod Kumar Kewat,Sarika Kushwaha 대한수학회 2018 대한수학회보 Vol.55 No.1
Let $R_{u^2, v^2, w^2, p}$ be a finite non chain ring $ \F_p[u,v,w]\langle u^2,$ $v^2, w^2$, $uv-vu, vw-wv, uw-wu \rangle$, where $p$ is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u^2, v^2, w^2, p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length $8n$ over $\F_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.
Quantum codes with improved minimum distance
Emre Kolotouglu,Mustafa Sari 대한수학회 2019 대한수학회보 Vol.56 No.3
The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order $q$ are presented by La Garcia in \cite{Guardia}. Being inspired by La Garcia's the paper, here we extend the results over a finite field with $q^2$ elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of $q^2$-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in \cite{Guardia}.
The Construction and Viterbi Decoding of New (2k, k, l) Convolutional Codes
Peng, Wanquan,Zhang, Chengchang Korea Information Processing Society 2014 Journal of information processing systems Vol.10 No.1
The free distance of (n, k, l) convolutional codes has some connection with the memory length, which depends on not only l but also on k. To efficiently obtain a large memory length, we have constructed a new class of (2k, k, l) convolutional codes by (2k, k) block codes and (2, 1, l) convolutional codes, and its encoder and generation function are also given in this paper. With the help of some matrix modules, we designed a single structure Viterbi decoder with a parallel capability, obtained a unified and efficient decoding model for (2k, k, l) convolutional codes, and then give a description of the decoding process in detail. By observing the survivor path memory in a matrix viewer, and testing the role of the max module, we implemented a simulation with (2k, k, l) convolutional codes. The results show that many of them are better than conventional (2, 1, l) convolutional codes.