A Class of Left Dihedral Codes Over Rings $mathbb{F}_q+umathbb{F}_q$

Yuan CAO; Yonglin CAO; Jian GAO

doi:10.1587/transfun.E100.A.2585

A Class of Left Dihedral Codes Over Rings $mathbb{F}_q+umathbb{F}_q$

Yuan CAO, Yonglin CAO, Jian GAO

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Let $mathbb{F}_q$ be a finite field of q elements, $R=mathbb{F}_q+umathbb{F}_q$ (u²=0) and D_2n=<x, y | xⁿ=1, y²=1, yxy=x^-1> be a dihedral group of order n. Left ideals of the group ring R[D_2n] are called left dihedral codes over R of length 2n, and abbreviated as left D_2n-codes over R. Let n be a positive factor of q^e+1 for some positive integer e. In this paper, any left D_2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A_i and outer codes C_i, where A_i is a cyclic code over R of length n and C_i is a linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code C_i is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D_2n-code is determined and all self-dual left D_2n-codes over R are presented, respectively.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E100-A No.12 pp.2585-2593

Publication Date: 2017/12/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E100.A.2585

Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)

Category: Coding Theory and Techniques

Authors

Yuan CAO
  Shandong University of Technology
Yonglin CAO
  Shandong University of Technology
Jian GAO
  Shandong University of Technology

Keyword

left dihedral code, finite chain ring, self-dual code, LCD code

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Yuan CAO, Yonglin CAO, Jian GAO, "A Class of Left Dihedral Codes Over Rings $mathbb{F}_q+umathbb{F}_q$" in IEICE TRANSACTIONS on Fundamentals, vol. E100-A, no. 12, pp. 2585-2593, December 2017, doi: 10.1587/transfun.E100.A.2585.
Abstract: Let $mathbb{F}_q$ be a finite field of q elements, $R=mathbb{F}_q+umathbb{F}_q$ (u²=0) and D_2n=<x, y | xⁿ=1, y²=1, yxy=x^-1> be a dihedral group of order n. Left ideals of the group ring R[D_2n] are called left dihedral codes over R of length 2n, and abbreviated as left D_2n-codes over R. Let n be a positive factor of q^e+1 for some positive integer e. In this paper, any left D_2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A_i and outer codes C_i, where A_i is a cyclic code over R of length n and C_i is a linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code C_i is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D_2n-code is determined and all self-dual left D_2n-codes over R are presented, respectively.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/transfun.E100.A.2585/_p

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@ARTICLE{e100-a_12_2585,
author={Yuan CAO, Yonglin CAO, Jian GAO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Class of Left Dihedral Codes Over Rings $mathbb{F}_q+umathbb{F}_q$},
year={2017},
volume={E100-A},
number={12},
pages={2585-2593},
abstract={Let $mathbb{F}_q$ be a finite field of q elements, $R=mathbb{F}_q+umathbb{F}_q$ (u²=0) and D_2n=<x, y | xⁿ=1, y²=1, yxy=x^-1> be a dihedral group of order n. Left ideals of the group ring R[D_2n] are called left dihedral codes over R of length 2n, and abbreviated as left D_2n-codes over R. Let n be a positive factor of q^e+1 for some positive integer e. In this paper, any left D_2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A_i and outer codes C_i, where A_i is a cyclic code over R of length n and C_i is a linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code C_i is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D_2n-code is determined and all self-dual left D_2n-codes over R are presented, respectively.},
keywords={},
doi={10.1587/transfun.E100.A.2585},
ISSN={1745-1337},
month={December},}

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TY - JOUR
TI - A Class of Left Dihedral Codes Over Rings $mathbb{F}_q+umathbb{F}_q$
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2585
EP - 2593
AU - Yuan CAO
AU - Yonglin CAO
AU - Jian GAO
PY - 2017
DO - 10.1587/transfun.E100.A.2585
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E100-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2017
AB - Let $mathbb{F}_q$ be a finite field of q elements, $R=mathbb{F}_q+umathbb{F}_q$ (u²=0) and D_2n=<x, y | xⁿ=1, y²=1, yxy=x^-1> be a dihedral group of order n. Left ideals of the group ring R[D_2n] are called left dihedral codes over R of length 2n, and abbreviated as left D_2n-codes over R. Let n be a positive factor of q^e+1 for some positive integer e. In this paper, any left D_2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A_i and outer codes C_i, where A_i is a cyclic code over R of length n and C_i is a linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code C_i is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D_2n-code is determined and all self-dual left D_2n-codes over R are presented, respectively.
ER -