• Journal of Communications and Networks
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• v.16 no.3
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• pp.270-279
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• 2014

The Viterbi decoding algorithm, which provides maximum - likelihood decoding, is currently considered the most widely used technique for the decoding of codes having a state description, including the class of linear error-correcting convolutional codes. Two classes of nonbinary convolutional codes are presented. Distance preserving mapping convolutional codes and M-ary convolutional codes are designed, respectively, from the distance-preserving mappings technique and the implementation of the conventional convolutional codes in Galois fields of order higher than two. We also investigated the performance of these codes when combined with a multiple frequency-shift keying (M-FSK) modulation scheme to correct narrowband interference (NBI) in power-line communications channel. Themodification of certain detectors of the M-FSK demodulator to refine the selection and the detection at the decoder is also presented. M-FSK detectors used in our simulations are discussed, and their chosen values are justified. Interesting and promising obtained results have shown a very strong link between the designed codes and the selected detector for M-FSK modulation. An important improvement in gain for certain values of the modified detectors was also observed. The paper also shows that the newly designed codes outperform the conventional convolutional codes in a NBI environment.

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.915-923
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• 2015

Self-dual codes have been actively studied because of their connections with other mathematical areas including t-designs, invariant theory, group theory, lattices, and modular forms. We presented the building-up construction for self-dual codes over GF(q) with $q{\equiv}1$ (mod 4), and over other certain rings (see [19], [20]). Since then, the existence of the building-up construction for the open case over GF(q) with $q=p^r{\equiv}3$ (mod 4) with an odd prime p satisfying $p{\equiv}3$ (mod 4) with r odd has not been solved. In this paper, we answer it positively by presenting the building-up construction explicitly. As examples, we present new optimal self-dual [16, 8, 7] codes over GF(7) and new self-dual codes over GF(7) with the best known parameters [24, 12, 9].

• The Transactions of the Korean Institute of Electrical Engineers
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• v.37 no.7
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• pp.498-509
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• 1988

A BF(breadth-first) linear quadtree as a new data structure for image data is suggested, which enables us to compress the image data efficiently and to make operations of the compressed data easily. It is a list of path names for black nodes as the linear quadtree is. The path name for each black node of a BF linear quadtree is represented as a sequence of path codes from the root node to itself, whereas that of linear quadtree as a sequence of path codes from the root node to itself and fill characters for cut-offed path from it to any n-level node which corresponds to a pixel of an image. The BF linear quadtree provides a more efficent compression ratio than the linear quadtree does, because the former does not require redundant characters, fill characters, for the cut-offed paths. Several operations for image processing can be also implemented on this hierarchical structure efficiently, because it is composed of only the black nodes ad the linear quadtree is . In this paper, algorithms for several operations on the BF linear quadtree are defined and analyzed. Experimental results for forur image data are also given and discussed.

• Proceedings of the IEEK Conference
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• 2000.07a
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• pp.298-301
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• 2000

We discuss the iterative methods for linear systems on a single processing node of the HITACHI SR8000. Each processing node of the SR8000 is a shared memory parallel computer which is composed of eight RISC processors with a pseudo-vector facility. We implement highly optimized codes for basic linear operations including a matrix-vector product and apply them to the conjugate gradient (CG) and the conjugate residual (CR) methods for linear systems. Our tuned codes for both method score nearly 50% of the theoretical peak performance, which is the best in the sense that it corresponds to an asymptotic performance of the inner product.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1627-1638
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• 2018

In this paper, we study an special type of cyclic codes called skew cyclic codes over the ring ${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$, where p is a prime number. This set of codes are the result of module (or ring) structure of the skew polynomial ring (${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$)[$x;{\theta}$] where $v^3=1$ and ${\theta}$ is an ${\mathbb{F}}_p$-automorphism such that ${\theta}(v)=v^2$. We show that when n is even, these codes are either principal or generated by two elements. The generator and parity check matrix are proposed. Some examples of linear codes with optimum Hamming distance are also provided.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.33-45
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• 2008

Lee weight is more appropriate for some practical situations than Hamming weight as it takes into account magnitude of each digit of the word. In this paper, we obtain a sufficient condition over the number of parity check digits for codes correcting random errors and simultaneously detecting burst errors with Lee weight consideration.

• Communications of Mathematical Education
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• v.6
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• pp.371-381
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• 1997

Let p be an odd prime number. We introduce simple and useful decoding algorithm for orthogonal Latin square codes of order p. Let H be the parity check matrix of orthogonal Latin square code. For any x ${\in}$ GF(p)$^{n}$, we call xH$^{T}$ the syndrome of x. This method is based on the syndrome decoding for linear codes. In L$_{p}$, we need to find the first and the second coordinates of codeword in order to correct the errored received vector.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.10 no.7
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• pp.3353-3370
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• 2016

The way of combination of Product Perfect Codes (PPCs) is based on the theory of short codes constructing long codes. PPCs have larger embedding rate than Hamming codes by expending embedding columns in a coding block, and they have been proven to enhance the performance of the F₅ steganographic method. In this paper, the proposed modified product codes called MPCs are introduced as an efficient way to embed more data than PPCs by increasing 2^r₂-1-r₂ embedding columns. Unlike PPC, the generation of the check matrix H in MPC is random, and it is different from PPC. In addition a simple solving way of the linear algebraic equations is applied to figure out the problem of expending embedding columns or compensating cases. Furthermore, the MPCs over ℤ₄ have been proposed to further enhance not only the performance but also the computation speed which reaches O(n₁+σ). Finally, the proposed ℤ₄-MPC intends to maximize the embedding rate with maintaining less distortion , and the performance surpasses the existing improved product perfect codes. The performance of large embedding rate should have the significance in the high-capacity of covert communication.

• Journal of Communications and Networks
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• v.10 no.3
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• pp.253-257
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• 2008

In this paper, the performance of generic orthogonal space-time block codes (OSTBCs) introduced by Alamouti [2], Tarokh [3], and Su and Xia [11] is analyzed. We first define one-dimensional component symbol error function (ODSEF) from the exact expression of the pairwise error probability of an OSTBC. Utilizing the ODSEF and the bit error probability (BEP) expression for quadrature amplitude modulation (QAM) introduced by Cho and Yoon [9], the exact closed-form expressions for the BEP of linear OSTBCs with QAM in quasi-static Rayleigh fading channel are derived. We also derive the exact closed-form of the BEP for some OSTBCs which have at least one message symbol transmitted with unequal power via all transmit antennas.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.443-460
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• 2023

A binary linear code is said to be a ℤ₂-double cyclic code if its coordinates can be partitioned into two subsets such that any simultaneous cyclic shift of the coordinates of the subsets leaves the code invariant. These codes were introduced in [6]. A ℤ₂-double cyclic code is called reversible if reversing the order of the coordinates of the two subsets leaves the code invariant. In this note, we give necessary and sufficient conditions for a ℤ₂-double cyclic code to be reversible. We also give a relation between reversible ℤ₂-double cyclic code and LCD ℤ₂-double cyclic code for the separable case and we present a few examples to show that such a relation doesn't hold in the non-separable case. Furthermore, we list examples of reversible ℤ₂-double cyclic codes of length ≤ 10.