• 대한수학회지
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• 제51권4호
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• pp.853-866
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• 2014

Constacyclic codes of length $p^s$ over $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ are precisely the ideals of the ring $\frac{R[x]}{<x^{p^s}-1>}$. In this paper, we investigate constacyclic codes of length $p^s$ over R. The units of the ring R are of the forms ${\gamma}$, ${\alpha}+u{\beta}$, ${\alpha}+u{\beta}+u^2{\gamma}$ and ${\alpha}+u^2{\gamma}$, where ${\alpha}$, ${\beta}$ and ${\gamma}$ are nonzero elements of $\mathbb{F}_{p^m}$. We obtain the structures and Hamming distances of all (${\alpha}+u{\beta}$)-constacyclic codes and (${\alpha}+u{\beta}+u^2{\gamma}$)-constacyclic codes of length $p^s$ over R. Furthermore, we classify all cyclic codes of length $p^s$ over R, and by using the ring isomorphism we characterize ${\gamma}$-constacyclic codes of length $p^s$ over R.

• 대한수학회지
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• 제58권3호
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• pp.571-595
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• 2021

In this paper we study the algebraic structure of ℤ_pℤ_p[u]/k>-cyclic codes, where u^k = 0 and p is a prime. A ℤ_pℤ_p[u]/k>-linear code of length (r + s) is an R_k-submodule of ℤ^r_p × R^s_k with respect to a suitable scalar multiplication, where R_k = ℤ_p[u]/k>. Such a code can also be viewed as an R_k-submodule of ℤ_p[x]/r - 1> × R_k[x]/s - 1>. A new Gray map has been defined on ℤ_p[u]/k>. We have considered two cases for studying the algebraic structure of ℤ_pℤ_p[u]/k>-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤ_pℤ_p[u]/k>-linear codes. Examples have been given to construct ℤ_pℤ_p[u]/k>-cyclic codes, through which we get codes over ℤ_p using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.

• 대한수학회보
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• 제56권5호
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• pp.1235-1255
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• 2019

In this paper, we introduce SR-additive codes as a generalization of the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes, where S is an R-algebra and an SR-additive code is an R-submodule of $S^{\alpha}{\times}R^{\beta}$. In particular, the definitions of bilinear forms, weight functions and Gray maps on the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes are generalized to SR-additive codes. Also the singleton bound for SR-additive codes and some results on one weight SR-additive codes are given. Among other important results, we obtain the structure of SR-additive cyclic codes. As some results of the theory, the structure of cyclic ${\mathbb{Z}}_2{\mathbb{Z}}_4$, ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$, ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$ and $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$-additive codes are presented.

• 대한수학회보
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• 제55권1호
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• pp.115-137
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• 2018

Let $R_{u{^2},v^2,w^2,p}$ be a finite non chain ring ${\mathbb{F}}_p[u,v,w]{\langle}u^2,\;v^2,\;w^2,\;uv-vu,\;vw-wv,\;uw-wu{\rangle}$, where p is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u{^2},v^2,w^2,p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length 8n over ${\mathbb{F}}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.

• 대한수학회보
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• 제49권1호
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• pp.135-143
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• 2012

We present two kinds of construction methods for self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$. Specially, the second construction (respectively, the first one) preserves the types of codes, that is, the constructed codes from Type II (respectively, Type IV) is also Type II (respectively, Type IV). Every Type II (respectively, Type IV) code over $\mathbb{F}_2+u\mathbb{F}_2$ of free rank larger than three (respectively, one) can be obtained via the second construction (respectively, the first one). Using these constructions, we update the information on self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$ of length 9 and 10, in terms of the highest minimum (Hamming, Lee, or Euclidean) weight and the number of inequivalent codes with the highest minimum weight.

• 대한수학회보
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• 제55권4호
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• pp.1189-1208
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• 2018

The aim of this paper is to study the class of ${\Lambda}$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal{R}}_a=\frac{{\mathbb{F}_{p^m}}[u]}{{\langle}u^a{\rangle}}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+{\cdots}+u^{a-1}{\mathbb{F}}_{p^m}$, for all units ${\Lambda}$ of ${\mathcal{R}}_a$ that have the form ${\Lambda}={\Lambda}_0+u{\Lambda}_1+{\cdots}+u^{a-1}{\Lambda}_{a-1}$, where ${\Lambda}_0,{\Lambda}_1,{\cdots},{\Lambda}_{a-1}{\in}{\mathbb{F}}_{p^m}$, ${\Lambda}_0{\neq}0$, ${\Lambda}_1{\neq}0$. The algebraic structure of all ${\Lambda}$-constacyclic codes of length $2p^s$ over ${\mathcal{R}}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.

• 대한수학회보
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• 제50권5호
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• pp.1513-1521
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• 2013

In this paper we will study cyclic codes over some special rings: $\mathbb{F}_q[u]/(u^i)$, $\mathbb{F}_q[u_1,{\ldots},u_i]/(u^2_1,u^2_2,{\ldots},u^2_i,u_1u_2-u_2u_1,{\ldots},u_ku_j-u_ju_k,{\ldots})$, and $\mathbb{F}_q[u,v]/(u^i,v^j,uv-vu)$, where $\mathbb{F}_q$ is a field with $q$ elements $q=p^r$ for some prime number $p$ and $r{\in}\mathbb{N}-\{0\}$.

• 대한수학회보
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• 제56권6호
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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.

• 대한수학회논문집
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• 제39권1호
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• pp.31-43
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• 2024

Let $R_e\,=\,\frac{{\mathbb{F}}_{p^m}[u]}{{\langle}u^e{\rangle}}$, where p is a prime number, e is a positive integer and u^e = 0. In this paper, we first characterize the structure of cyclic codes of length p^s over R_e. These codes will be classified into 2^e distinct types. Among other results, in the case that e = 4, the torsion codes of cyclic codes of length p^s over R₄ are obtained. Also, we present some examples of cyclic codes of length p^s over R_e.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.159-167
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• 2006

The transform domain characterization of cyclic codes over finite fields using Discrete Fourier Transform(DFT) over an appropriate extension field is well known. In this paper, we extend this transform domain characterization for cyclic codes over $F_p + uF_p +... + u^{k-l}F_p$. We give a way to characterize cyclic codes over $F_p + uF_p +... + u^{k-l}F_p$ by Mattson-Solomon polynomials and multiple defining sets.