• Korean Journal of Mathematics
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• v.32 no.3
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• pp.485-496
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• 2024

In this study, we examine ℓ-quasi-cyclic self-dual codes of length ℓm over 𝔽₂, provided that the polynomial X^m - 1 has exactly four distinct irreducible factors in 𝔽₂[X]. We find the standard form of generator matrices of codes over the ring R ≅ 𝔽_q[X]/(X^m - 1) and the conditions for the codes to be self-dual. We explicitly determine the forms of generator matrices of self-dual codes of lengths 2 and 4 over R.

• ETRI Journal
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• v.39 no.3
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• pp.319-325
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• 2017

The decoding process of a quasi-cyclic low-density parity check code requires a unique type of rotator. These rotators, called multi-size cyclic-shifters (MSCSs), rotate input data with various sizes, where the size is the amount of data to be rotated. This paper proposes a low-complexity MSCS structure for the case when the sizes have a nontrivial common divisor. By combining the strong points of two previous structures, the proposed structure achieves the smallest area. The experimental results show that the area reduction was more than 14.7% when the proposed structure was applied to IEEE 802.16e as an example.

• Journal of Communications and Networks
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• v.13 no.3
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• pp.205-210
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• 2011

This paper presents an approach to the construction of multiple-rate quasi-cyclic low-density parity-check (LDPC) codes. Parity-check matrices of the proposed codes consist of $q{\times}q$ square submatrices. The block rows and block columns of the parity-check matrix correspond to the hyperplanes (${\mu}$-fiats) and points in Euclidean geometries, respectively. By decomposing the ${\mu}$-fiats, we obtain LDPC codes of different code rates and a constant code length. The code performance is investigated in term of the bit error rate and compared with those of LDPC codes given in IEEE standards. Simulation results show that our codes perform very well and have low error floors over the additive white Gaussian noise channel.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.419-437
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• 2019

This paper investigates skew ${\Theta}-{\lambda}$-constacyclic codes over $R=F_0{\oplus}F_1{\oplus}{\cdots}{\oplus}F_{k-1}$, where $F{_i}^{\prime}s$ are finite fields. The structures of skew ${\lambda}$-constacyclic codes over finite commutative semi-simple rings and their duals are provided. Moreover, skew ${\lambda}$-constacyclic codes of arbitrary length are studied under a new definition. We also show that a skew cyclic code of arbitrary length over finite commutative semi-simple rings is equivalent to either a cyclic code over R or a quasi-cyclic code over R.

• ETRI Journal
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• v.45 no.3
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• pp.404-417
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• 2023

This paper proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of medium to high code rates that can be applied in cloud data storage systems, requiring better error correction capabilities. The novelty of this method lies in the construction of sparse base matrices, using a girth greater than 4 that can then be expanded with a lift factor to produce high code rate QC-LDPC codes. Investigations revealed that the proposed large-sized QC-LDPC codes with high code rates displayed low encoding complexities and provided a low bit error rate (BER) of 10^-10 at 3.5 dB E_b/N₀ than conventional LDPC codes, which showed a BER of 10^-7 at 3 dB E_b/N₀. Subsequently, implementation of the proposed QC-LDPC code in a softwaredefined radio, using the NI USRP 2920 hardware platform, was conducted. As a result, a BER of 10^-6 at 4.2 dB E_b/N₀ was achieved. Then, the performance of the proposed codes based on their encoding-decoding speeds and storage overhead was investigated when applied to a cloud data storage (GCP). Our results revealed that the proposed codes required much less time for encoding and decoding (of data files having a 10 MB size) and produced less storage overhead than the conventional LDPC and Reed-Solomon codes.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.41 no.12
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• pp.1745-1747
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• 2016

In this paper, we propose a construction method of irregular quasi-cyclic low-density parity-check codes based on perfect difference families with various block sizes. The proposed codes have advantages in that they support various values with respect to code rate, length, and degree distribution. Also, this construction enables very short lengths which are usually difficult to be achieved by a random construction. We verify via simulations the error-correcting performance of the proposed codes.

• Journal of Communications and Networks
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• v.17 no.2
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• pp.157-161
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• 2015

A new block low-density parity-check (Block-LDPC) code based on quadratic permutation polynomials (QPPs) is proposed. The parity-check matrix of the Block-LDPC code is composed of a group of permutation submatrices that correspond to QPPs. The scheme provides a large range of implementable LDPC codes. Indeed, the most popular quasi-cyclic LDPC (QC-LDPC) codes are just a subset of this scheme. Simulation results indicate that the proposed scheme can offer similar error performance and implementation complexity as the popular QC-LDPC codes.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.19 no.8
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• pp.1839-1844
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• 2015

The low-density parity-check(LDPC) code has been adopted in many communication standards due to its error correcting performance, and the quasi-cyclic LDPC(QC-LDPC) is widely used because of implementation easiness. In the QC-LDPC decoder, a cyclic-shifter is required to rotate data in various sizes. This kind of cyclic-shifters are called multi-size circular shifter(MSCS), and this paper proposes a low-complexity structure for MSCS. In the conventional serially-placed two barrel-rotators, the unnecessary multiplexers are revealed and removed, leading to low-complexity. The experimental results show that the area is reduced by about 12%.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.723-735
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• 2006

In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over $\mathbb{Z}_4$ is free if C is generated by a polynomial which divides $X^m-1$. In this paper, we prove that a necessary and sufficient condition for a cyclic code over $\mathbb{Z}_pk$ of length m to be free is that it is generated by a polynomial which divides $X^m-1$. We also show that this can be extended to finite local rings with a principal maximal ideal.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.7 no.10
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• pp.2411-2429
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• 2013

In this paper, we extend the case of information exchange error mitigation for the distributed orthogonal space-time block code (DOSTBC) for two transmit antennas to distributed quasi-orthogonal space-time block code (DQOSTBC) for four transmit antennas. A rate 1 full-diversity DQOSTBC for four transmit antennas is designed. The code matrix changes according to different information exchange error cases, so full diversity is maintained even if not all information exchange is correct. We also perform analysis of the pairwise error probability. The performance analysis indicates that the proposed rate 1 DQOSTBC outperforms rate 1/2 DOSTBC for four transmit antennas at the same transmission rate, which is confirmed by the simulation results.