• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1095-1110
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• 2017

The t-wise intersection is a useful property of a linear code due to its many applications. Recently, the second author determined the t-wise intersection of a relative two-weight code. By using this result and generalizing the finite projective geometry method, we will present the t-wise intersection of a relative three-weight code and its applications in this paper.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.29-44
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• 2021

The set D of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one-weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let D be a subset of ��2n, and W (resp. V ) be a subspace of ��2 (resp. ��2n). We define the linear code ��D(W; V ) with defining set D and restricted to W, V by $${\mathcal{C}}_D(W;V )=\{(s+u{\cdot}x)_{x{\in}D^{\ast}}|s{\in}W,u{\in}V\}$$. We obtain two- or three-weight codes by taking D to be a Vasil'ev code of length n = 2m - 1(m ≥ 3) and a suitable choices of W. We do the same job for D being the complement of a Vasil'ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.731-736
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• 2017

It is well-known that there exists a constant-weight $[s{\theta}_{k-1},k,sq^{k-1}]_q$ code for any positive integer s, which is an s-fold simplex code, where ${\theta}_j=(q^{j+1}-1)/(q-1)$. This gives an upper bound $n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+n_q(k,d)$ for any positive integer d, where $n_q(k,d)$ is the minimum length n for which an $[n,k,d]_q$ code exists. We construct a two-weight $[s{\theta}_{k-1}+1,k,sq^{k-1}]_q$ code for $1{\leq}s{\leq}k-3$, which gives a better upper bound $n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+1+n_q(k-1,d)$ for $1{\leq}d{\leq}q^s$. As another application, we prove that $n_q(5,d)={\sum_{i=0}^{4}}{\lceil}d/q^i{\rceil}$ for $q^4+1{\leq}d{\leq}q^4+q$ for any prime power q.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.107-120
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• 2008

There has been a recent growth of interest in codes with respect to a newly defined non-Hamming metric grown as the Rosenbloom-Tsfasman metric (RT, or $\rho$, in short). In this paper, the definitions of the Lee complete $\rho$ weight enumerator and the exact complete $\rho$ weight enumerator of a code over $M_n_\times_s(Z_4)$ are given, and the MacWilliams identities with respect to this RT metric for the two weight enumerators of a linear code over $M_n_\times_s(Z_4)$ are proven too. At last, we also prove that the MacWilliams identities for the Lee and exact complete $\rho$ weight enumerators of a linear code over $M_n_\times_s(Z_4)$ are the generalizations of the MacWilliams identities for the Lee and complete weight enumerators of the corresponding code over $Z_4$.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.493-505
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• 2000

We discuss the class of projective systems whose supports are the complement of the union of two linear subspaces in general position. We express the weight enumerators of the codes generated by these projective systems using two simplex codes corresponding to given linear subspaces. We also prove these codes are uniquely determined upto equivalence by their weight enumerators.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.14 no.6
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• pp.590-603
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• 1989

In tnis paper, the CRC-CCITT code and primitive polynomial CRC code are selected for analysing error detecting performance. However, general formulas for obtaining the weight distribution of these two CRC codes are not so far dericed. So, a new method for calculating the weight distribution of the shortened cyclic Hamming code is presented and an undetected error probability of these two codes is obtained when used in cell of ATM for broadband ISDN user-network interface. Consequently, we show that CRC code too much does affect its error detection performance. All the computer simulation is performed by IBM PC/AT.

• Current Optics and Photonics
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• v.3 no.6
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• pp.531-540
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• 2019

Spectral Amplitude Coding Optical Code Division Multiple Access (SAC OCDMA) is an advanced technique in asynchronous environments. This paper proposes design and implementation of a novel quantum logic gate (QLG) code, with code construction algorithm generated without following any code mapping procedures for SAC system. The proposed code has a unitary matrices property with maximum overlap of one chip for various clients and no overlaps in spectra for the rest of the subscribers. Results indicate that a single algorithm produces the same length increment for codes with weight greater than two and follows the same signal to noise ratio (SNR) and bit error rate (BER) calculations for a higher number of users. This paper further examines the performance of a QLG code based SAC-OCDMA system with NAND and direct detection techniques. BER analysis was carried out for the proposed code and results were compared with existing MDW, RD and GMP codes. We demonstrate that the QLG code based system performs better in terms of cardinality, which is followed by improved BER. Numerical analysis reveals that for error free transmission (10^-9), the suggested code supports approximately 170 users with code weight 4. Our results also conclude that the proposed code provides improvement in the code construction, cross-correlation and minimization of noises.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.32 no.4C
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• pp.370-378
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• 2007

The most appropriate low density parity check (LDPC) code for hybrid automatic repeat request (HARQ) system suitable for future multimedia communication systems is presented in this paper. HARQ system with punctured LDPC code is investigated at first. And two transmission schemes with parallel concatenated LDPC code are also presented and their performances are analyzed according to the various values of mean column weight (MCW). As a result, the parallel concatenated LDPC code with the diversity effect of information bit is considered to be more appropriate for HARQ system considering the throughput as well as error performance.

• Proceedings of the IEEK Conference
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• 2000.11a
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• pp.65-68
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• 2000

In CDMA, the currently used PN code acquisition methods detect the phase of the incoming PN signal on the basis of ML estimation. In serial search strategy, it takes relatively long time for PN code acquisition. A study related to reduce processing time of PN code acquisition is important. Therefore we propose a reference code design for a new algorithm that a reference code and the received PN code are fully correlated to detect the transmitted PN code, in detail, the relative phase offset j of c⒡. After choosing two weight factors, we show the mathematical analysis and derivation for the reference code design.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.17 no.8
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• pp.2157-2169
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• 2023

In order to handle the problem that puncture patterns will change the position distribution of original information bits and frozen bits in polar codes, which affects performance of puncture polar codes further, a minimum row weight and polar spectrum based puncture polar codes construction algorithm (called PA-MRWP) is proposed in this paper. The algorithm calculates row weight of generator matrix and sorts the row weight in ascending order first. Next, the positions with the minimum row weight are selected as initial puncture positions. If the rows with the same row weight cannot all be punctured, polar spectrum based auxiliary puncture scheme is used. In sub-channels with the same row weight, rows corresponding to the polarized sub-channels with higher reliability are selected as puncture positions to construct puncture vector, and the reliability is calculated based on polar spectrum. It is actually a two-step selection strategy, the proposed minimum row weight puncture (MRWP) algorithm is used for primary selection and polar spectrum based auxiliary puncture is used for adjustment. Simulation results show that, compared with worst quality puncture (WQP) algorithm, the proposed PA-MRWP algorithm and Gaussian approximation-aided minimum row weight puncture (GA-MRWP) algorithm provide gains of about 0.46 dB and 0.29 dB at bit error rate (BER) of 10^-4, respectively when code length N=400, code rate R=1/2. In addition, the proposed puncture algorithms improve the BER performance significantly with respect to quasi-uniform puncture (QUP) algorithm.