• Korean Journal of Mathematics
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• v.17 no.1
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• pp.43-52
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• 2009

The Galois ring is a finite extension of the ring of integers modulo a prime power. We consider characters on Galois rings. In analogy with finite fields, we investigate complete Gauss sums over Galois rings. In particular, we analyze [1, Proposition 3] and give some lemmas related to [1, Proposition 3].

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.519-535
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• 2018

Let ${\mathcal{R}}$ denote the Galois ring of characteristic $p^n$, where p is a prime. In this paper, we investigate the elementary properties of Gauss sums over ${\mathcal{R}}$ in accordance with conditions of characters of Galois rings, and we restate results for Gauss sums in [1, 2, 3, 7, 12, 13]. Also, we compute the modulus of the Gauss sums.

• Journal of Applied and Pure Mathematics
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• v.4 no.3_4
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• pp.211-219
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• 2022

We study MDS(maximum distance separable) self-dual codes over Galois ring R = GR(2^m, r). We prove that there exists an MDS self-dual code over R of length n if (n - 1) divides (2^r - 1), and 2^m divides n. We also provide the current state of the problem for the existence of MDS self-dual codes over Galois rings.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.751-764
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• 2016

We investigate the self-dual codes over Galois rings and determine the mass formula for self-dual codes over Galois rings $GR(p^2,2)$.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.571-583
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• 2020

The Galois ring R of characteristic pⁿ having p^mn elements is a finite extension of the ring of integers modulo pⁿ, where p is a prime number and n, m are positive integers. In this paper, we develop the concepts of Jacobi sums over R and under the assumption that the generating additive character of R is trivial on maximal ideal of R, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.

• East Asian mathematical journal
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• v.38 no.5
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• pp.583-591
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• 2022

Let K be an imaginary quadratic field and 𝒪 be an order in K. Let H_𝒪 be the ring class field of 𝒪. Furthermore, for a positive integer N, let K_𝒪,N be the ray class field modulo N𝒪 of 𝒪. When the discriminant of 𝒪 is different from -3 and -4, we construct an extended form class group which is isomorphic to the Galois group Gal(K_𝒪,N/H_𝒪) and describe its Galois action on K_𝒪,N in a concrete way.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.2
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• pp.255-270
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• 2020

We studied the MDS self-dual codes over finite chain rings. We stated the projection and lifting of codes over the finite chain rings with respect to the MDS self-dual codes, and then we applied the results to the MDS self-dual codes over Galois rings.

• Journal for History of Mathematics
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• v.19 no.3
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• pp.85-94
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• 2006

We review Galois rings and permutation polynomials over Galois rings with emphasis on cryptological properties of Dickson polynomials and conclude that it is not plausible to obtain the Dickson scheme which has a Galois ring as a plaintext space.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.767-777
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• 2019

We present a new way of obtaining the complete factorization of $X^n-1$ for n = 15, 17, 19 over the 4-adic ring ${\mathcal{O}}_4[X]$ of integers and thus over the Galois rings $GR(2^e,2)$. As a result, we determine all cyclic codes of lengths 15, 17 and 19 over those rings. This extends our previous work on such cyclic codes of odd lengths less than 15.