• Kyungpook Mathematical Journal
• /
• v.56 no.2
• /
• pp.349-355
• /
• 2016

For any integer $n{\geq}2$, each palindrome of n induces a circulant graph of order n. It is known that for each integer $n{\geq}2$, there is a one-to-one correspondence between the set of (resp. aperiodic) palindromes of n and the set of (resp. connected) circulant graphs of order n (cf. [2]). This bijection gives a one-to-one correspondence of the palindromes ${\sigma}$ with $gcd({\sigma})=1$ to the connected circulant graphs. It was also shown that the number of palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ is the same number of aperiodic palindromes of n. Let $a_n$ (resp. $b_n$) be the number of aperiodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). Let $c_n$ (resp. $d_n$) be the number of periodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). In this paper, we calculate the numbers $a_n$, $b_n$, $c_n$, $d_n$ in two ways. In Theorem 2.3, we $n_d$ recurrence relations for $a_n$, $b_n$, $c_n$, $d_n$ in terms of $a_d$ for $d{\mid}n$ and $d{\neq}n$. Afterwards, we nd formulae for $a_n$, $b_n$, $c_n$, $d_n$ explicitly in Theorem 2.5.

• Journal of KIISE:Computer Systems and Theory
• /
• v.26 no.8
• /
• pp.999-1007
• /
• 1999

이 논문은 재귀원형군 G(2^m , 2^k )를 그래프 이론적 관점에서 고찰하고 정점이 서로소인 사이클과 그래프 invariant에 관한 위상 특성을 제시한다. 재귀원형군은 1 에서 제안된 다중 컴퓨터의 연결망 구조이다. 재귀원형군 {{{{G(2^m , 2^k )가 길이 사이클을 가질 필요 충분 조건을 구하고, 이 조건하에서 G(2^m , 2^k )는 가능한 최대 개수의 정점이 서로소이고 길이가l`인 사이클을 가짐을 보인다. 그리고 정점 및 에지 채색, 최대 클릭, 독립 집합 및 정점 커버에 대한 그래프 invariant를 분석한다.Abstract In this paper, we investigate recursive circulant G(2^m , 2^k ) from the graph theory point of view and present topological properties of G(2^m , 2^k ) concerned with vertex-disjoint cycles and graph invariants. Recursive circulant is an interconnection structure for multicomputer networks proposed in 1 . A necessary and sufficient condition for recursive circulant {{{{G(2^m , 2^k ) to have a cycle of lengthl` is derived. Under the condition, we show that G(2^m , 2^k ) has the maximum possible number of vertex-disjoint cycles of length l`. We analyze graph invariants on vertex and edge coloring, maximum clique, independent set and vertex cover.

• Journal of the Korean Institute of Telematics and Electronics
• /
• v.21 no.4
• /
• pp.7-12
• /
• 1984

A composite - discrete courier transform (CDFT) is developed, which can diagonalize a real symmetric circulant matrix. In general the circulant matrices can be diagonalized by the discrete Fourier transform (DFT). With the analysis of the variance distributions of the DFT and CDFT for the general symmetric covariance matrix of real signals, the DFT and CDFT are compared with respect to the rate distortion performance measure. The results show that the CDFT is more efficient than the DFT in bit rate reduction. In addition, for a particular 64$\times$64 points covariance matrix (f(q)=(0.95)q), the amount of the relative average bit rate reduction for the CDFT with respect to the DFT is obtained by 0.0095 bit with a numerical calculation.

• Journal of Korea Multimedia Society
• /
• v.16 no.7
• /
• pp.897-904
• /
• 2013

The recursive circulant network G(N,d) can be widely used in the design and implementation of parallel processing architectures. It consists of N identical nodes, each node is connected through bidirectional, point-to-point communication channels to different neighbors by jumping $d^i$, where $0{\leq}i{\leq}{\lceil}{\log}_dN{\rceil}$ - 1. In this paper, we investigate the routing of a message on $G(2^m,4)$, a special kind of RCN, that is key to the performance of this network. On $G(2^m,4)$ we would like to transmit k packets from a source node to k destination nodes simultaneously along paths on this network, the $i^{th}$ packet will be transmitted along the $i^{th}$ path, where $1{\leq}k{\leq}m-1$, $0{{\leq}}i{{\leq}}m-1$. In order for all packets to arrive at a destination node quickly and securely, we present an $O(m^4)$ routing algorithm on $G(2^m,4)$ for generating a set of one-to-many node-disjoint and nearly shortest paths, where each path is either shortest or nearly shortest and the total length of these paths is nearly minimum since the path is mainly determined by employing the Hungarian method.

• Journal of the Institute of Electronics Engineers of Korea TC
• /
• v.45 no.7
• /
• pp.1-8
• /
• 2008

Low Density Parity Check codes(LDPC) are recently focused on communication systems due to its good performance. The standard of WiBro has also included LDPC codes as a channel coding. The weak point of implementation for LDPC encoder is that conventional binary Matrix Vector Multiplier has many clock cycles which limit throughput. In this paper, we propose semi-parallel architecture by using cyclic shift registers and exclusive-OR without conventional Matrix Vector Multipliers over the standard parity check matrices with Circulant Permutation Matrices(CPM). Furthermore, multi-rate encoder is designed by using proposed architecture. Our encoder with multi-rate for IEEE 802.16e LDPC has lower clock cycles and higher throughput.

• Kyungpook Mathematical Journal
• /
• v.14 no.1
• /
• pp.97-100
• /
• 1974

• Proceedings of the IEEK Conference
• /
• 1998.10a
• /
• pp.471-474
• /
• 1998

In this paper, we consider the problem of optimal broadcasting in recursive circulants under multi-port communication model. Recursive circulant G(N, d) that is defined to be a circulant graph with N vertices and jumps of powers of d is a useful interconnection network from the viewpoint of network metrices. Our model assumes that a processor can transmit a message to $\alpha$ neighboring processors simultaneously where $\alpha$ is two or three. For the broadcasting problem, we introduce 3-trees and 4-trees. And then we show that 3-trees and 4-trees are minimum broadcast trees in 2-port model and 3-port model. Using the above results, we show that recursive circulants g(2m, 2) have optimum broadcasting time in 2-port model and 3-port model.

• Journal of applied mathematics & informatics
• /
• v.29 no.5_6
• /
• pp.1571-1581
• /
• 2011

In this paper, we construct inequivalent Hadamard matrices based on several new and old full orthogonal designs, using circulant and symmetric block matrices. Not all orthogonal designs produce inequivalent Hadamard matrices, because the corresponding systems of equations do not possess solutions. The systems of equations arising when we search for inequivalent Hadamard matrices from full orthogonal designs using circulant and symmetric block matrices, can be concisely described using the periodic autocorrelation function of the generators of the block matrices. We use Maple, Magma, C and Unix tools to find many new inequivalent Hadamard matrices.

• The Journal of the Korea institute of electronic communication sciences
• /
• v.9 no.4
• /
• pp.403-408
• /
• 2014

In this paper, we consider the encoding scheme of low density codes. In particular, we propose the fast encoding algorithm of the low density codes using circulant permutation matrices, The fast encoding algorithm is similar to the fast Hadamard transform, and the results show that the proposed scheme is a simple and fast encoding algorithm of the low density codes.

• Journal of KIISE:Computer Systems and Theory
• /
• v.31 no.1_2
• /
• pp.19-26
• /
• 2004

In this paper, we investigate the longest fault-free paths joining every pair of vertices in a double loop network with faulty vertices and/or edges, and show that a bipartite double loop network G(mn；1, m) is strongly hamiltonian-laceable when the number of faulty elements is two or less. G(mn；1, m) is bipartite if and only if m is odd and n is even.