• 대한수학회보
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• 제54권1호
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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.

• 대한수학회보
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• 제52권6호
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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].

• 대한수학회보
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• 제53권6호
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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.

• 대한수학회보
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• 제55권1호
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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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• 제34권5_6호
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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.

• 대한수학회지
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• 제58권3호
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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.

• 한국정보과학회논문지:소프트웨어및응용
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• 제32권3호
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• pp.141-152
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• 2005

본 논문에서는 새로운 감성기반 영상검색방법을 제안한다. 서로 다른 색상, 명도, 도트크기를 나타내는 30개의 랜덤 패턴이 제시될 때 인간이 느끼는13가지 감성("like", "beautiful", "natural", "dynamic", "warm", "gay", "cheerful", "unstable", "light", "strong", "gaudy", "hard", "heavy") 평가 데이타로부터 질의 칼라코드와 질의 그레이코드로 명명한 질의 감성 표시자를 설계한다. 감성영상검색을 위해서 질의 감성을 선택하면 질의를 표현하는 칼라코드와 그레이코드가 선택되고 데이타베이스의 영상의 색상 정보를 나타내는 DB 칼라코드와 명도와 도트크기 정보를 나타내는 DB그레이코드값을 추출하여, 칼라코드간의 매칭과 그레이 코드간의 매칭을 통해 유사도를 판단한다. 또한 검색과정에 사용자의 의도를 반영하여 질의 칼라코드와 질의 그레이코드사이의 가중치와 칼라코드내의 가중치를 자동적으로 갱신하는 새로운 유사도 피드백 방법을 제안한다. 430개의 영상에 대해 실험한 결과 최초 질의에 대해 적합한 영상이 부적합한 영상보다 많았으며 유사도 피드백을 사용함에 따라 적합한 영상의 개수가 증가하였다.

• Journal of electromagnetic engineering and science
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• 제4권1호
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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.

• 한국산학기술학회논문지
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• 제5권5호
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• pp.409-412
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• 2004

본 논문에서는 감소된 복잡도를 가진 Turbo TCM(Turbo Trellis Coded Modulation) 구조를 제안하고 그 성능을 분석한다. Turbo Codes와 대역폭 효율을 증대시키기 위한 변조기술을 적용한 부호화기를 구성하고 수신기에서는 이진 Turbo Codes와 유사한 형태지만 다소 상이한 심벌 대 심벌 MAP 반복복호를 사용한다. 반복복호에 따른 성능 분석을 했으며 제안된 구조는 Gray mapping을 이용한 Turbo Codes와 비교한 결과 $BER=10^{-2}$인 경우 약 2.5dB 우수한 이득을 얻음을 알 수 있다.

• 대한수학회지
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• 제59권1호
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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.