• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.971-997
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• 2018

The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime p are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^n$, $n{\geq}1$ over the quaternary field $F_4$ are obtained. These codes are observed to be subcodes of Q codes of length $p^n$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a Q code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic Q codes of order 6 exist.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.25-43
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• 2005

Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length $2^{n}$ generated by these idempotents are completely described. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n-1 idempotents are also obtained. The case n = 2,3 is dealt separately.

• Journal of applied mathematics & informatics
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• v.36 no.5_6
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• pp.359-368
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• 2018

In this paper, we investigate quasi-cyclic codes over the ring $R={\mathbb{F}}_2+v{\mathbb{F}}_2$, where $v^2=v$. We investigate the structure of generators for one-generator quasi-cyclic codes over R and their minimal spanning sets. Moreover, we find the rank and a lower bound on minimum distances of free quasi-cyclic codes over R. Further, we find a relationship between cyclic codes over a different ring and quasi-cyclic codes of index 2 over R.

• Journal of Institute of Control, Robotics and Systems
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• v.8 no.4
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• pp.299-302
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• 2002

Error detection and correction using CRC and the general class of cyclic codes is an important part of designing reliable data transmission schemes. The decoding method for cyclic codes using covering patterns is easily-implementable, and its complexity de-pends on the number of covering patterns employed. Determination of the minimal set of covering patterns for a given code is an open problem. In this paper, an efficient search method for constructing minimal sets of covering patterns is proposed and compared with several existing search methods. The result is applicable to various codes of practical interest.

• 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.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.38A no.2
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• pp.156-163
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• 2013

The use of an error-correcting code is essential in communication systems where the channel is noisy. When channel coding parameters are unknown at a receiver side, decoding becomes difficult. To perform decoding without the channel coding information, we should estimate the parameters. In this paper, we introduce a method to reconstruct the generator polynomial of BCH(Bose-Chaudhuri-Hocquenghem) codes based on the idea that the generator polynomial is compose of minimal polynomials and BCH code is cyclic code. We present a probability compensation method to improve the reconstruction performance. This is based on the concept that a random data pattern can also be divisible by a minimal polynomial of the generator polynomial. And we confirm the performance improvement through an intensive computer simulation.

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.773-793
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• 2001

The explicit expressions for the 2n+1 primitive idempotents in R/sub pⁿ/ = F[x]/< x/sup pⁿ/ -1>, where F is the field of prime power order q and the multiplicative order of q modulo pⁿ is ø(pⁿ)/2(n≥1 and p is an odd prime), are obtained. An algorithm for computing the generating polynomials of the minimal QR cyclic codes of length pⁿ, generated by these primitive idempotents, is given and hence some bounds on the minimum distance of some QR codes of prime length over GF(q)(q=2, 3, ...) are obtained.

• Steel and Composite Structures
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• v.33 no.1
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• pp.1-18
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• 2019

In the present study, the behavior of steel plate shear walls (SPSW) with variable column flexural stiffness is experimentally and numerically investigated. Altogether six one-bay one-story specimens, three moment resisting frames (MRFs) and three SPSWs, were designed, fabricated and tested. Column flexural stiffness of the first specimen pair (one MRF and one SPSW) corresponded to the value required by the design codes, while for the second and third pair it was reduced by 18% and 36%, respectively. The quasi-static cyclic test result indicate that SPSW with reduced column flexural stiffness have satisfactory performance up to 4% story drift ratio, allow development of the tension field over the entire infill panel, and cause negligible column "pull-in" deformation which indicates that prescribed minimal column flexural stiffness value, according to AISC 341-10, might be conservative. In addition, finite element (FE) pushover simulations using shell elements were developed. Such FE models can predict SPSW cyclic behavior reasonably well and can be used to conduct numerical parametric analyses. It should be mentioned that these FE models were not able to reproduce column "pull-in" deformation indicating the need for further development of FE simulations with cyclic load introduction which will be part of another paper.

• Steel and Composite Structures
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• v.33 no.1
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• pp.19-36
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• 2019

Cyclic behaviour of composite (steel-concrete) plate shear walls (CPSW) with variable column flexural stiffness is experimentally and numerically investigated. The investigation included design, fabrication and testing of three pairs of one-bay one-storey CPSW specimens. The reference specimen pair was designed in way that its column flexural stiffness corresponds to the value required by the design codes, while within the other two specimen pairs column flexural stiffness was reduced by 18% and 36%, respectively. Specimens were subjected to quasi-static cyclic tests. Obtained results indicate that column flexural stiffness reduction in CPSW does not have negative impact on the overall behaviour allowing for satisfactory performance for up to 4% storey drift ratio while also enabling inelastic buckling of the infill steel plate. Additionally, in comparison to similar steel plate shear wall (SPSW) specimens, column "pull-in" deformations are less pronounced within CPSW specimens. Therefore, the results indicate that prescribed minimal column flexural stiffness value used for CPSW might be conservative, and can additionally be reduced when compared to the prescribed value for SPSWs. Furthermore, finite element (FE) pushover simulations were conducted using shell and solid elements. Such FE models can adequately simulate cyclic behaviour of CPSW and as such could be further used for numerical parametric analyses. It is necessary to mention that the implemented pushover FE models were not able to adequately reproduce column "pull-in" deformation and that further development of FE simulations is required where cyclic loading of the shear walls needs to be simulated.