• Korean Journal of Mathematics
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• v.17 no.4
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• pp.385-397
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• 2009

We study codes over the polynomial ring ${\mathbb{F}}_q[D]$ and their projections to the finite rings ${\mathbb{F}}_q[D]/(D^m)$ and the weight enumerators of self-dual codes over these rings. We also give the formula for the number of codewords of minimum weight in the projections.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.703-715
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• 2021

There are several methods for constructing self-dual codes. Among them, the building-up construction is a powerful method. Recently, Kim and Choi proposed special building-up constructions which use symmetric generator matrices for self-dual codes over GF(q), where q is odd. In this paper, we study the same method when q is even.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.37-50
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• 2020

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.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.965-989
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• 2015

Let p, ${\ell}$ be distinct odd primes, q be an odd prime power with gcd(q, p) = gcd(q,${\ell}$) = 1, and m, n be positive integers. In this paper, we determine all self-dual, self-orthogonal and complementary-dual negacyclic codes of length $2p^{n{\ell}m}$ over the finite field ${\mathbb{F}}_q$ with q elements. We also illustrate our results with some examples.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1357-1369
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• 2006

It is known that if C is an [24m + 2l, 12m + l, d] selfdual binary linear code with $0{\leq}l<11,\;then\;d{\leq}4m+4$. We present a sufficient condition for the nonexistence of extremal selfdual binary linear codes with d=4m+4,l=1,2,3,5. From the sufficient condition, we calculate m's which correspond to the nonexistence of some extremal self-dual binary linear codes. In particular, we prove that there are infinitely many such m's. We also give similar results for additive self-dual codes over GF(4) of length n=6m+1.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.3
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• pp.181-194
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• 2023

Let GR(2^m, r) be a Galois ring with even characteristic. We are interested in the existence of MDS(Maximum Distance Separable) self-dual codes over GR(2^m, r). In this paper, we prove that there exists an MDS self-dual code over GR(2^m, r) with parameters [n, n/2, n/2 + 1] if (n - 1) | (2^r - 1) and 8 | n.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1385-1422
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• 2019

Let p be an odd prime. The algebraic structure of all negacyclic codes of length $8_{p^s}$ over the finite commutative chain ring ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ where $u^2=0$ is studied in this paper. Moreover, we classify all negacyclic codes of length $8_{p^s}$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ into 5 cases, i.e., $p^m{\equiv}1$ (mod 16), $p^m{\equiv}3$, 11 (mod 16), $p^m{\equiv}5$, 13 (mod 16), $p^m{\equiv}7$, 15 (mod 16) and $p^m{\equiv}9$ (mod 16). From that, the structures of dual and some self-dual negacyclic codes and number of codewords of negacyclic codes are obtained.

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

• Journal of Institute of Control, Robotics and Systems
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• v.14 no.7
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• pp.647-655
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• 2008

This paper presents real-time algorithm for an ultrasonic Local Positioning System(LPS). An ultrasonic LPS consists of 4 transmitters and n receivers. Each transmitter transmits an sequential ultrasonic signal to avoid interference of ultrasonic signal. This method is a noneffective application for a fast object. Because receiver detects four sequential transmissive ultrasonic signal and calculates a position. This paper proposes the method which 4 transmitters transmit simultaneous ultrasonic signal and each transmitter distinguished by frequencies and codes. And Auto-Correlation Function(ACF) method separates codes from an ultrasonic echo signal which is interference of each transmitter's code. If the receiver uses only ACF method, it is difficult to implement real time application for increased computation. This paper implements LPS using dual frequencies and ACF method. Using dual frequencies reduces codes length. The reduced codes length save computation in ACF. To prove this algorithm by experiment, high performance DSP(digital signal processor) used. The result shows the performance of the designed system is good enough positioning.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.30 no.1C
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• pp.1-8
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• 2005

In this paper, we present a novel LDPC code construction called as semi-algebra low density parity check(LDPC) codes which is one kind of deterministic LDPC code based on dual-diagonal sub-matrix. The constructing method results in a class of high rate LDPC codes. Codes in this class have a large girth and good minimum distances. Furthermore, they can be implemented by simple parallel array architecture using cyclic shift register and perform well with the iterative decoding.