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

• Computers and Concrete
• /
• v.2 no.1
• /
• pp.31-53
• /
• 2005

In the present structural codes the safety verification is based on a linear analysis of the structure and the satisfaction of ultimate and serviceability limit states, using a semi-probabilistic security format through the consideration of partial safety factors, which affect the action values and the characteristic values of the material properties. In this context, if a non-linear structural analysis is wanted a difficulty arises, because the global safety coefficient, which could be obtained in a straightforward way from the non-linear analysis, is not directly relatable to the different safety coefficient values usually used for the different materials, as is the case for reinforced concrete structures. The work here presented aims to overcome this difficulty by proposing a methodology that generalises the format of safety verification based on partial safety factors, well established in structural codes within the scope of linear analysis, for cases where non-linear analysis is needed. The methodology preserves the principal assumptions made in the codes as well as a reasonable simplicity in its use, including a realistic definition of the material properties and the structural behaviour, and it is based on the evaluation of a global safety coefficient. Some examples are presented aiming to clarify and synthesise all the options that were taken in the application of the proposed methodology, namely how to transpose the force distributions obtained with a non-linear analysis into design force distributions. One of the most important features of the proposed methodology, the ability for comparing the simplified procedures for second order effects evaluation prescribed in the structural codes, is also presented in a simple and systematic way. The potential of the methodology for the development and assessment of alternative and more accurate procedures to those already established in codes of practice, where non-linear effects must be considered, is also indicated.

• Korean Journal of Mathematics
• /
• v.15 no.1
• /
• pp.79-85
• /
• 2007

We study codes over the polynomial ring $\mathbb{F}_q[D]$ and introduce the notion of basic codes which play a fundamental role in the theory.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
• /
• 2022.05a
• /
• pp.365-366
• /
• 2022

The distributed code is a type of linear codes that can be used for coding and federated learning for privacy. In the distributed code, privacy or confidential information is not dependent to each other because the information of each code is not included with other codes. In this paper, we examine the properties of these distributed codes and present techniques for synthesizing new sets of distributed codes from previously known distributed codes. In addition, we propose several scenarios in which combined codes can be used.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.33 no.12C
• /
• pp.1062-1067
• /
• 2008

ABBA codes, a class of quasi-orthognal space-time block codes (QoSTBC) proposed by Tirkkonen and others, allow full rate and a fast maximum likelihood (ML) decoding, but do not have full diversity. In this paper, a linear complex precoder is proposed for ABBA codes to achieve full rate and full diversity. Moreover, the same diversity produce as that of orthogonal space-time block code with linear complex precoder (OSTBC-LCP) is achieved. Meanwhile, the size of the linear complex precoder can be reduced by half without affecting performance, which means the same complexity of decoding as that of the conventional ABBA code is guaranteed.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.717-734
• /
• 2018

By choosing defining set properly, several classes of linear codes with a few weights over the finite field ${\mathbb{F}}_p$ are constructed for an odd prime p, and the complete weight enumerators of these classes of codes are determined.

• Journal of Communications and Networks
• /
• v.17 no.1
• /
• pp.12-20
• /
• 2015

Convolutional codes which are terminated by direct truncation (DT) and zero tail termination provide unequal error protection. When DT terminated convolutional codes are used to encode short messages, they have interesting error protection properties. Such codes match the significance of the output bits of common quantizers and therefore lead to a low mean square error (MSE) when they are used to encode quantizer outputs which are transmitted via a noisy digital communication system. A code construction method that allows adapting the code to the channel is introduced, which is based on time-varying convolutional codes. We can show by simulations that DT terminated convolutional codes lead to a lower MSE than standard block codes for all channel conditions. Furthermore, we develop an MSE approximation which is based on an upper bound on the error probability per information bit. By means of this MSE approximation, we compare the convolutional codes to linear unequal error protection code construction methods from the literature for code dimensions which are relevant in analog to digital conversion systems. In numerous situations, the DT terminated convolutional codes have the lowest MSE among all codes.

• Journal of the Korean Mathematical Society
• /
• v.47 no.6
• /
• pp.1167-1182
• /
• 2010

We study codes meeting a generalized version of the Singleton bound for higher weights. We show that some of the higher weight enumerators of these codes are uniquely determined. We give the higher weight enumerators for MDS codes, the Simplex codes, the Hamming codes, the first order Reed-Muller codes and their dual codes. For the putative [72, 36, 16] code we find the i-th higher weight enumerators for i = 12 to 36. Additionally, we give a version of the generalized Singleton bound for non-linear codes.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.36 no.10C
• /
• pp.605-613
• /
• 2011

In most digital communication systems over the noisy channel, some form of forward error correction scheme is employed for reliable communications. If one wants to recover the transmitted message without any knowledge of the error correcting codes employed, it is of utmost importance to figure out and reconstruct the error correcting codes. In this paper, we propose two algorithms of reconstructing linear cyclic codes from the corrupted received bit sequence, one for general linear binary cyclic codes and the other for Reed-Solomon codes. For two algorithms, we ran computer simulations and the performances are shown to be superior to those with the conventional LWM method.

• Journal of the Korean Mathematical Society
• /
• v.58 no.3
• /
• pp.571-595
• /
• 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.