• Korean Journal of Mathematics
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• 제22권4호
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• pp.725-742
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• 2014

In this paper, we find all inequivalent classes of self-orthogonal codes over $Z_{p^2}$ of lengths $l{\leq}3$ for all primes p, using similar method as in [3]. We find that the classification of self-orthogonal codes over $Z_{p^2}$ includes the classification of all codes over $Z_p$. Consequently, we classify all the codes over $Z_p$ and self-orthogonal codes over $Z_{p^2}$ of lengths $l{\leq}3$ according to the automorphism group of each code.

• 대한수학회지
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• 제59권1호
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• pp.193-204
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• 2022

We present construction methods for free self-orthogonal (self-dual or Type II) codes over ℤ₄[v]/〈v² + 2v〉 which is one of the finite commutative local non-chain Frobenius rings of order 16. By considering their Gray images on ℤ₄, we give a construct method for a code over ℤ₄. We have some new and optimal codes over ℤ₄ with respect to the minimum Lee weight or minimum Euclidean weight.

• Korean Journal of Mathematics
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• 제25권2호
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• pp.201-209
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• 2017

There exists already mass formula which is the number of self orthogonal codes in $GF(q)^n$, but not proof of it. In this paper we described some theories about finite geometry and by using them proved the mass formula when $q=p^m$, p is odd prime.

• 한국전자통신학회논문지
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• 제10권11호
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• pp.1239-1244
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• 2015

본 논문에서는 길쌈 Doubly 직교 부호의 최적 스팬을 찾기 위한 새로운 방법(Convolutional Self-Doubly Orthogonal, CDO)을 제안한다. 이 새로운 방법은 병렬 Implicitly-Exhaustive 탐색방법을 사용하는데, 이 방법으로 R =1/2 CDO 코드에 대한 최적의 스팬을 찾기 위해서 계산시간을 감소시키는 방법으로 동적 검색 공간 감소 방법을 적용했다. 제안된 알고리듬을 모의실험한 결과 기존의 방법에 비해서 계산시간이 감소되었고, 오류 정정 성능이 향상되었음을 확인하였다.

• 대한수학회지
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• 제52권5호
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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.

• 대한수학회지
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• 제39권2호
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• pp.289-317
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• 2002

We associate binary codes to polynomials over fields of characteristic two and show that the binary Golay codes are associated to the Mathieu group polynomials in characteristics two. We give two more polynomials whose Galois group in $M_{12}$ but different self-orthogonal binary codes are associated. Also, we find a family of $M_{24}$-coverings which includes previous ones.

• 대한수학회보
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• 제56권1호
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• pp.131-150
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• 2019

In this paper, we consider the structure of ${\gamma}$-constacyclic codes of length $2p^s$ over the Galois ring $GR(p^a,m)$ for any unit ${\gamma}$ of the form ${\xi}_0+p{\xi}_1+p^2z$, where $z{\in}GR(p^a,m)$ and ${\xi}_0$, ${\xi}_1$ are nonzero elements of the set ${\mathcal{T}}(p,m)$. Here ${\mathcal{T}}(p,m)$ denotes a complete set of representatives of the cosets ${\frac{GR(p^a,m)}{pGR(p^a,m)}}={\mathbb{F}}p^m$ in $GR(p^a,m)$. When ${\gamma}$ is not a square, the rings ${\mathcal{R}}_p(a,m,{\gamma})=\frac{GR(p^a,m)[x]}{{\langle}x^2p^s-{\gamma}{\rangle}}$ is a chain ring with maximal ideal ${\langle}x^2-{\delta}{\rangle}$, where ${\delta}p^s={\xi}_0$, and the number of codewords of ${\gamma}$-constacyclic code are provided. Furthermore, the self-orthogonal and self-dual ${\gamma}$-constacyclic codes of length $2p^s$ over $GR(p^a,m)$ are also established. Finally, we determine the Rosenbloom-Tsfasman (RT) distances and weight distributions of all such codes.