ON LCD CODES OVER FINITE CHAIN RINGS

Durgun, Yilmaz Korean Mathematical Society 2020 대한수학회보 Vol.57 No.1

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. LCD cyclic codes have been known as reversible cyclic codes that had applications in data storage. Due to a newly discovered application in cryptography, interest in LCD codes has increased again. Although LCD codes over finite fields have been extensively studied so far, little work has been done on LCD codes over chain rings. In this paper, we are interested in structure of LCD codes over chain rings. We show that LCD codes over chain rings are free codes. We provide some necessary and sufficient conditions for an LCD code C over finite chain rings in terms of projections of linear codes. We also showed the existence of asymptotically good LCD codes over finite chain rings.

On LCD codes over finite chain rings

Ylmaz Durgun 대한수학회 2020 대한수학회보 Vol.57 No.1

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. LCD cyclic codes have been known as reversible cyclic codes that had applications in data storage. Due to a newly discovered application in cryptography, interest in LCD codes has increased again. Although LCD codes over finite fields have been extensively studied so far, little work has been done on LCD codes over chain rings. In this paper, we are interested in structure of LCD codes over chain rings. We show that LCD codes over chain rings are free codes. We provide some necessary and sufficient conditions for an LCD code $C$ over finite chain rings in terms of projections of linear codes. We also showed the existence of asymptotically good LCD codes over finite chain rings.

핸드라이트 숫자 인식을 위한 조합형 체인코드 생성

박종안(Jong-an Park),강성관(Seong-kwan Kang),박용준(Yong-jun Park) 한국정보기술학회 2013 Proceedings of KIIT Conference Vol.2013 No.5

본 연구에서는 디지털 기기에 핸드라이트로 입력되는 숫자를 인식하기 위한 새로운 조합형 체인코드를 제안하였다. 제안한 조합형 체인코드는 일반 체인코드의 방향 민감성에 기인한 오류를 방지하기 위하여 핸드라이트 선성분의 다중 체인코드를 조합하여 새로운 코드 인덱스를 설정하는 방식을 이용한다. 이것은 중복 샘플링 되는 숫자 선성분 내에서 일정 구간의 체인코드를 확인하여 4단계 인덱스를 제공함으로서 연산 량을 감소시키면서도 국부적 인식 오류를 감소시키는 방향성 핸드라이트 인식방식이다. 이것은 간단한 시뮬레이션 결과로 그 타당성을 확인할 수 있었다. In this study, we propose the new combined chain code for recognizing the numbers being input with the handwrite to the digital devices. The proposed Combined chain code uses the technique to set up a new code index combined a multi-Chain code of the handwrite ray components in order to avoid errors due to the sensitivity of the direction of the normal chain code. It is the directional handwrite recognition technique that reduce the amount of computation and the local recognition error as providing the four-step index through checking chain code of certain sections within the duplicate sampled the number line components. It was able to verify the validity through the result of a simple simulation.

Color Image Coding Based on Shape-Adaptive All Phase Biorthogonal Transform

( Xiaoyan Wang ),( Chengyou Wang ),( Xiao Zhou ),( Zhiqiang Yang ) 한국정보처리학회 2017 Journal of information processing systems Vol.13 No.1

This paper proposes a color image coding algorithm based on shape-adaptive all phase biorthogonal transform (SA-APBT). This algorithm is implemented through four procedures: color space conversion, image segmentation, shape coding, and texture coding. Region-of-interest (ROI) and background area are obtained by image segmentation. Shape coding uses chain code. The texture coding of the ROI is prior to the background area. SA-APBT and uniform quantization are adopted in texture coding. Compared with the color image coding algorithm based on shape-adaptive discrete cosine transform (SA-DCT) at the same bit rates, experimental results on test color images reveal that the objective quality and subjective effects of the reconstructed images using the proposed algorithm are better, especially at low bit rates. Moreover, the complexity of the proposed algorithm is reduced because of uniform quantization.

응용프로그램 역분석 방지를 위한 코드블록 암호화 방법

정동우(Dong-Woo Jung),김형식(Hyong-Shik Kim),박중길(Joong Gil Park) 한국정보보호학회 2008 정보보호학회논문지 Vol.18 No.2

실행코드의 변조와 역분석(reverse engineering)을 방지하기 위한 대표적인 방법은 실행코드를 암호화하는 것이다. 본 논문에서는 키체인(key chaining) 방식의 블록암호화 기법을 이용하여 응용프로그램을 암호화하는 방법을 제안한다. 키체인 방식의 블록암호화 기법은 키가 블록의 내부에 은닉되어 있고 각 블록의 키가 서로 다르다는 장점을 갖지만, 제어이동을 필요로 하는 프로그램에 적용하기에는 적합하지 않다고 알려져 있다. 본 논문에서는 실행코드에서의 제어이동 명령어에 대해서도 키체인 방식을 효과적으로 적용할 수 있도록 블록을 변형시키거나 중복시키는 방법을 제시하고, MIPS 명령어집합을 이용하여 가능성을 분석한다. One of the typical methods to prevent tampering and reverse engineering on executable codes is to encrypt them. This paper proposes a code block cipher method based on key chaining to encrypt the code. The block cipher by key chaining has been known to be inadequate for encrypting the code with control transfer, even though the key chaining has advantage of hiding the keys in blocks and making the individual keys different from block to block. This paper proposes a block transformation and duplication method to apply the block cipher by key chaining to the executable codes with control transfer instructions, and shows the idea works with the MIPS instruction set.

SR-ADDITIVE CODES

Mahmoudi, Saadoun,Samei, Karim Korean Mathematical Society 2019 대한수학회보 Vol.56 No.5

In this paper, we introduce SR-additive codes as a generalization of the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes, where S is an R-algebra and an SR-additive code is an R-submodule of $S^{\alpha}{\times}R^{\beta}$. In particular, the definitions of bilinear forms, weight functions and Gray maps on the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes are generalized to SR-additive codes. Also the singleton bound for SR-additive codes and some results on one weight SR-additive codes are given. Among other important results, we obtain the structure of SR-additive cyclic codes. As some results of the theory, the structure of cyclic ${\mathbb{Z}}_2{\mathbb{Z}}_4$, ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$, ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$ and $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$-additive codes are presented.

$SR$-additive codes

Saadoun Mahmoudi,Karim Samei 대한수학회 2019 대한수학회보 Vol.56 No.5

In this paper, we introduce $SR$-additive codes as a generalization of the classes of $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$ and $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, where $S$ is an $R$-algebra and an $SR$-additive code is an $R$-submodule of $S^{\alpha}\times R^{\beta}$. In particular, the definitions of bilinear forms, weight functions and Gray maps on the classes of $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$ and $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are generalized to $SR$-additive codes. Also the singleton bound for $SR$-additive codes and some results on one weight $SR$-additive codes are given. Among other important results, we obtain the structure of $SR$-additive cyclic codes. As some results of the theory, the structure of cyclic $\mathbb{Z}_{2}\mathbb{Z}_{4}$, $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$, $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, $(\mathbb{Z}_{2})(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + u^{2}\mathbb{Z}_{2})$, $(\mathbb{Z}_{2} + u\mathbb{Z}_{2} )(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + u^{2}\mathbb{Z}_{2})$, $(\mathbb{Z}_{2})(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + v\mathbb{Z}_{2})$ and $(\mathbb{Z}_{2} + u\mathbb{Z}_{2} )(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + v\mathbb{Z}_{2})$-additive codes are presented.

Color Image Coding Based on Shape-Adaptive All Phase Biorthogonal Transform

Wang, Xiaoyan,Wang, Chengyou,Zhou, Xiao,Yang, Zhiqiang Korea Information Processing Society 2017 Journal of information processing systems Vol.13 No.1

This paper proposes a color image coding algorithm based on shape-adaptive all phase biorthogonal transform (SA-APBT). This algorithm is implemented through four procedures: color space conversion, image segmentation, shape coding, and texture coding. Region-of-interest (ROI) and background area are obtained by image segmentation. Shape coding uses chain code. The texture coding of the ROI is prior to the background area. SA-APBT and uniform quantization are adopted in texture coding. Compared with the color image coding algorithm based on shape-adaptive discrete cosine transform (SA-DCT) at the same bit rates, experimental results on test color images reveal that the objective quality and subjective effects of the reconstructed images using the proposed algorithm are better, especially at low bit rates. Moreover, the complexity of the proposed algorithm is reduced because of uniform quantization.

On a class of constacyclic codes of length $2p^s$ over $\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}$

Hai Q. Dinh,Bac Trong Nguyen,Songsak Sriboonchitta 대한수학회 2018 대한수학회보 Vol.55 No.4

The aim of this paper is to study the class of $\Lambda$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal R}_a=\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}=\mathbb F_{p^m} + u \mathbb F_{p^m}+ \dots + u^{a-1}\mathbb F_{p^m}$, for all units $\Lambda$ of $\mathcal R_a$ that have the form $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{p^m}$, $\Lambda_0 \,{\not=}\, 0, \, \Lambda_1 \,{\not=}\, 0$. The algebraic structure of all $\Lambda$-constacyclic codes of length $2p^s$ over ${\mathcal R}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.

ON A CLASS OF CONSTACYCLIC CODES OF LENGTH 2p^s OVER $\frac{\mathbb{F}_{p^m}[u]}{{\langle}u^a{\rangle}}$

Dinh, Hai Q.,Nguyen, Bac Trong,Sriboonchitta, Songsak Korean Mathematical Society 2018 대한수학회보 Vol.55 No.4

The aim of this paper is to study the class of ${\Lambda}$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal{R}}_a=\frac{{\mathbb{F}_{p^m}}[u]}{{\langle}u^a{\rangle}}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+{\cdots}+u^{a-1}{\mathbb{F}}_{p^m}$, for all units ${\Lambda}$ of ${\mathcal{R}}_a$ that have the form ${\Lambda}={\Lambda}_0+u{\Lambda}_1+{\cdots}+u^{a-1}{\Lambda}_{a-1}$, where ${\Lambda}_0,{\Lambda}_1,{\cdots},{\Lambda}_{a-1}{\in}{\mathbb{F}}_{p^m}$, ${\Lambda}_0{\neq}0$, ${\Lambda}_1{\neq}0$. The algebraic structure of all ${\Lambda}$-constacyclic codes of length $2p^s$ over ${\mathcal{R}}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.