• ETRI Journal
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• 제29권3호
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• pp.381-389
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• 2007

In this paper we propose a graph-theoretic method based on linear congruence for constructing low-density parity check (LDPC) codes. In this method, we design a connection graph with three kinds of special paths to ensure that the Tanner graph of the parity check matrix mapped from the connection graph is without short cycles. The new construction method results in a class of (3, ${\rho}$)-regular quasi-cyclic LDPC codes with a girth of 12. Based on the structure of the parity check matrix, the lower bound on the minimum distance of the codes is found. The simulation studies of several proposed LDPC codes demonstrate powerful bit-error-rate performance with iterative decoding in additive white Gaussian noise channels.

• 한국통신학회논문지
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• 제31권9C호
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• pp.846-852
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• 2006

${\kappa}$-오류 선형복잡도는 통신 시스템 및 스트림 암호 시스템 등에 사용되는 수열의 안정성 여부를 판단하는 중요한 척도이다. 본 논문은 p가 소수이고 2가 모듈로 $p^2$의 원시근일 때 $p^m$-주기 이진 수열의 k-오류 선형복잡도와 해당 오류벡터를 효과적으로 구할 수 있는 알고리듬을 소개한다. 또한 암호학적인 관점에서 정의된 ${\kappa}$-오류 선형 복잡도의 의미를 부호 이론의 관점에서 살펴봄으로써 부호어의 길이가 $p^m$인 이진 순환 부호를 효과적으로 복호할 수 있는 알고리듬을 소개하며 이러한 부호의 최소 거리에 관한 중요한 성질들을 유도한다.

• 대한수학회지
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• 제43권6호
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• pp.1357-1369
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• 2006

It is known that if C is an [24m + 2l, 12m + l, d] selfdual binary linear code with $0{\leq}l<11,\;then\;d{\leq}4m+4$. We present a sufficient condition for the nonexistence of extremal selfdual binary linear codes with d=4m+4,l=1,2,3,5. From the sufficient condition, we calculate m's which correspond to the nonexistence of some extremal self-dual binary linear codes. In particular, we prove that there are infinitely many such m's. We also give similar results for additive self-dual codes over GF(4) of length n=6m+1.

• 한국통신학회논문지
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• 제32권10A호
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• pp.958-964
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• 2007

이 논문은 maximum a posteriori (MAP) 비트 검출(bit detection)의 비트 오류 확률 (bit error probability: BEP)과 비트 최소 평균 제곱 오류(bit minimum mean square error: bit MMSE)사이의 관계를 유도한다. BEP는 bit MMSE의 1/4 보다 크고 1/2보다 작음을 유도한다. 이 결론을 이용하면 bit-linear linear-dispersion (BLLD) 부호를 적용한 다중 입출력 (multiple-input multiple-output: MIMO) 통신 시스템에서 가우시안 채널의 mutual information의 미분 값의 하한과 상한을 BEP로부터 얻을 수 있고 나아가서 mutual information의 하한과 상한을 구할 수 있다.

• 대한수학회지
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• 제46권5호
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• pp.967-977
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• 2009

There are three standard weight functions on a linear code viz. Hamming weight, Lee weight, and Euclidean weight. Euclidean weight function is useful in connection with the lattice constructions [2] where the minimum norm of vectors in the lattice is related to the minimum Euclidean weight of the code. In this paper, we obtain an upper bound over the number of parity check digits for Euclidean weight codes detecting and correcting burst errors.

• 대한수학회보
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• 제58권5호
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• pp.1163-1173
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• 2021

In this paper, we obtain the MacWilliams identity for linear codes over finite chain rings with respect to homogeneous weight, and the product of chain rings.

• 한국정보보호학회:학술대회논문집
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• 한국정보보호학회 1996년도 종합학술발표회논문집
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• pp.286-294
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• 1996

In a remarkable paper 〔3〕, Hammons et al. showed that, when properly defined, the binary nonlinear Preparata code can be considered as the Gray map of a linear code eve. Z$_4$, the so-called Preparata code eve. Z$_4$. Recently, Yang and Helleseth 〔12〕 considered the generalized Hamming weights d$\_$r/(m) for Preparata codes of length 2$\^$m/ over Z$_4$ and exactly determined d$\_$r/, for r = 0.5,1.0,1.5,2,2.5 and 3.0. In particular, they completely determined d$\_$r/(m) for any r in the case of m $\leq$ 6. In this paper we show that the Preparata code of length 16 over Z$_4$ does not satisfy the chain condition.

• 대한수학회보
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• 제56권1호
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• pp.111-130
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• 2019

The concepts of pseudocodeword and pseudoweight play a fundamental role in the finite-length analysis of LDPC codes. The pseudoredundancy of a binary linear code is defined as the minimum number of rows in a parity-check matrix such that the corresponding minimum pseudoweight equals its minimum Hamming distance. By using the value assignment of Chen and Kløve we present new results on the pseudocodeword redundancy of binary linear codes. In particular, we give several upper bounds on the pseudoredundancies of certain codes with repeated and added coordinates and of certain shortened subcodes. We also investigate several kinds of k-dimensional binary codes and compute their exact pseudocodeword redundancy.

• 대한수학회보
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• 제60권3호
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• pp.829-844
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• 2023

In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders 2, 3, 4, and 5, and by applying the construction over the binary field and the ring F₂ + uF₂, we obtain extremal binary self-dual codes of various lengths: 12, 16, 20, 24, 32, 40, and 48. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code [24, 12, 8] and the unique Extended Quadratic Residue [48, 24, 12] Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in [10] by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.

• 대한수학회지
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• 제48권3호
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• pp.475-486
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• 2011

The paper deals with repeated low-density burst error detecting codes with a specied weight or less. Linear codes capable of detecting such errors have been studied. Further codes capable of correcting and simultaneously detecting such errors have also been dealt with. The paper obtains lower and upper bounds on the number of parity-check digits required for such codes. An example of such a code has also been provided.