• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.29-42
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• 2017

In this work, four circulant and quadratic double circulant (QDC) constructions are applied to the family of the rings $R_{k,m}$. Self-dual binary codes are obtained as the Gray images of self-dual QDC codes over $R_{k,m}$. Extremal binary self-dual codes of length 64 are obtained as Gray images of ${\lambda}-four$ circulant codes over $R_{2,1}$ and $R_{2,2}$. Extremal binary self-dual codes of lengths 66 and 68 are constructed by applying extension theorems to the ${\mathbb{F}}_2$ and $R_{2,1}$ images of these codes. More precisely, 10 new codes of length 66 and 39 new codes of length 68 are discovered. The codes with these weight enumerators are constructed for the first time in literature. The results are tabulated.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.2011-2023
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• 2015

This paper determines the structures of one-homogeneous weight codes over finite chain rings and studies the algebraic properties of these codes. We present explicit constructions of one-homogeneous weight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residue field, we obtain a family of optimal one-Hamming weight codes over the residue field. Further, we propose a generalized method that also includes the examples of optimal codes obtained by Shi et al. in [17].

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1617-1628
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• 2016

In this paper, we extend the results given in [3] to a nonchain ring $R_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_p$, where $v^p=v$ and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring $R_p$ and study the structure of their duals. We classify cyclic codes containing their duals over $R_p$ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map ${\pi}$ defined in [4], we give the parameters of the quantum codes of length pn over ${\mathbb{F}}_p$ which are obtained from cyclic codes over $R_p$. Finally, we illustrate the results by giving some examples.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.115-137
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• 2018

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.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.397-404
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• 2016

In this work, a method is given to construct quantum codes from cyclic codes over F₄ + vF₄ which will be denoted as R throughout the paper, where v² = v and a Gray map is defined between R and where F₄ is the field with 4 elements. Some optimal quantum code parameters and others will be presented at the end of the paper.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.571-595
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• 2021

In this paper we study the algebraic structure of ℤ_pℤ_p[u]/k>-cyclic codes, where u^k = 0 and p is a prime. A ℤ_pℤ_p[u]/k>-linear code of length (r + s) is an R_k-submodule of ℤ^r_p × R^s_k with respect to a suitable scalar multiplication, where R_k = ℤ_p[u]/k>. Such a code can also be viewed as an R_k-submodule of ℤ_p[x]/r - 1> × R_k[x]/s - 1>. A new Gray map has been defined on ℤ_p[u]/k>. We have considered two cases for studying the algebraic structure of ℤ_pℤ_p[u]/k>-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤ_pℤ_p[u]/k>-linear codes. Examples have been given to construct ℤ_pℤ_p[u]/k>-cyclic codes, through which we get codes over ℤ_p using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.

• Journal of KIISE:Software and Applications
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• v.32 no.3
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• pp.141-152
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• 2005

A new emotion-based image retrieval method is proposed in this paper. Query emotion descriptors called query color code and query gray code are designed based on the human evaluation on 13 emotions('like', 'beautiful', 'natural', 'dynamic', 'warm', 'gay', 'cheerful', 'unstable', 'light' 'strong', 'gaudy' 'hard', 'heavy') when 30 random patterns with different color, intensity, and dot sizes are presented. For emotion image retrieval, once a query emotion is selected, associated query color code and query gray code are selected. Next, DB color code and DB gray code that capture color and, intensify and dot size are extracted in each database image and a matching process between two color codes and between two gray codes are peformed to retrieve relevant emotion images. Also, a new relevance feedback method is proposed. The method incorporates human intention in the retrieval process by dynamically updating weights of the query and DB color codes and weights of an intra query color code. For the experiments over 450 images, the number of positive images was higher than that of negative images at the initial query and increased according to the relevance feedback.

• Journal of electromagnetic engineering and science
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• v.4 no.1
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• pp.17-22
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• 2004

In this paper, we implement an efficient MLC/PDL system for AWGN channels. In terms of the tradeoff between the hardware implementation and system performance, proposed algebra LDPC codes are optimized by the Gaussian approximation(GA) according to the rate of each level assigned by the capacity rule and chosen as the component code. System performance with Ungerboeck Partitioning(UP), Miked Partitioning(MP) and Gray Mapping(GM) of 8PSK are evaluated, respectively. Many results are presented in this paper; they can indicate that the proposed MLC/PDL system using optimized algebra LDPC codes with different code rate, capacity rule and Gray mapping(GM) can achieve the best performance.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.5 no.5
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• pp.409-412
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• 2004

This paper shows that the proposed Turbo TCM(Turbo Trellis Coded Modulation) has a good performance with a little complexity of decoder. The encoder structure, which is connected with Turbo Codes, is the proposed modulation technique for an efficient bandwidth, This method is used symbol by symbol MAP decoder of iteration similar to binary Turbo Codes in the receiver. The result shows that the BER performance according to iteration is improved about 2,5dB at $BER=10^{-2}$ compared to Turbo Codes with Gray mapping.

• Journal of the Korean Mathematical Society
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• v.59 no.1
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• pp.193-204
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• 2022

We present construction methods for free self-orthogonal (self-dual or Type II) codes over ℤ₄[v]/〈v² + 2v〉 which is one of the finite commutative local non-chain Frobenius rings of order 16. By considering their Gray images on ℤ₄, we give a construct method for a code over ℤ₄. We have some new and optimal codes over ℤ₄ with respect to the minimum Lee weight or minimum Euclidean weight.