American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020122

Codes over $ \frak m $-adic completion rings

Saadoun Mahmoudi and  Karim Samei ,

Department of Mathematics, Bu Ali Sina University, Hamedan, Iran

*Corresponding author

Received  April 2020 Revised  September 2020 Published  December 2020

The theory of linear codes over finite rings has been generalized to linear codes over infinite rings in two special cases; the ring of $ p $-adic integers and formal power series ring. These rings are examples of complete discrete valuation rings (CDVRs). In this paper, we generalize the theory of linear codes over the above two rings to linear codes over complete local principal ideal rings. In particular, we obtain the structure of linear and constacyclic codes over CDVRs. For this generalization, first we study linear codes over $ \hat{R}_{ \frak m} $, where $ R $ is a commutative Noetherian ring, $ \frak m = \langle \gamma\rangle $ is a maximal ideal of $ R $, and $ \hat{R}_{ \frak m} $ denotes the $ \frak m $-adic completion of $ R $. We call these codes, $ \frak m $-adic codes. Using the structure of $ \frak m $-adic codes, we present the structure of linear and constacyclic codes over complete local principal ideal rings.

Keywords: MDS code, constacyclic code, $ \frak m $-adic completion ring.
Mathematics Subject Classification: Primary: 11T71, 94B05; Secondary: , 13J10.
Citation: Saadoun Mahmoudi, Karim Samei. Codes over $ \frak m $-adic completion rings. Advances in Mathematics of Communications, doi: 10.3934/amc.2020122
References:
[1]

M. F. Atiyah and I. G. Macdonald, Intrduction to Commutative Algebra, University of Oxford, 1969. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[2]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.  Google Scholar

[3]

I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., 59 (1946), 54-106.  doi: 10.1090/S0002-9947-1946-0016094-3.  Google Scholar

[4]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.  Google Scholar

[5]

S. T. Dougherty, J. Kim and H. Kulosman, MDS codes over finite principal ideal rings, MDS Codes Over Finite Principal Ideal Rings, 50 (2009), 77-92.  doi: 10.1007/s10623-008-9215-5.  Google Scholar

[6]

S. T. Dougherty, S. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrete. Math., 305 (2005), 123-135.  doi: 10.1016/j.disc.2005.08.004.  Google Scholar

[7]

S. T. Dougherty, H. Liu and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ., 53 (2011), 39-53.   Google Scholar

[8]

S. T. Dougherty and H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia, 31B (2011), 331-343.  doi: 10.1016/S0252-9602(11)60233-6.  Google Scholar

[9]

S. T. Dougherty and Y. H. Park, Codes over the $p$-adic integers, Des. Codes Cryptogr., 39 (2006), 65-80.  doi: 10.1007/s10623-005-2542-x.  Google Scholar

[10]

D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[11]

E. E. Enochs and O. M. G. Jenda, Relative Homolodical Algebra, Walter de Gruyter, 2000. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[12]

K. Guenda and T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields Appl., 18 (2012), 1061-1075.  doi: 10.1016/j.ffa.2012.09.003.  Google Scholar

[13]

S. Jean-Pierre, Local Fields, Berlin, New York, 1980. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[14]

F. J. Makwilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[15]

H. Matsumura, Commutative Ring Theory, Cambridge, 1989. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[16]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[17]

K. R. McLean, Commutative Artinian principal ideal rings, Proc. London Math. Soc., 26 (1973), 249-272.  doi: 10.1112/plms/s3-26.2.249.  Google Scholar

[18]

K. Samei and S. Mahmoudi, Cyclic $R$-additive codes, Discrete Math., 340 (2017), 1657-1668.  doi: 10.1016/j.disc.2016.11.007.  Google Scholar

[19]

K. Samei and S. Mahmoudi, Singleton Bundes for $R$-additive codes, Adv. Math. Commun., 12 (2018), 107-114.  doi: 10.3934/amc.2018006.  Google Scholar

[20]

P. Solé, Open problems 2: Cyclic codes over rings and $p$-adic fields, in (G. Cohen and J. Wolfmann eds.) Coding Theory and Applications, Lect. Notes Comp. Sci., 338, Springer-Verlag, 1988. doi: 10.1007/BFb0019872.  Google Scholar

show all references

References:
[1]

M. F. Atiyah and I. G. Macdonald, Intrduction to Commutative Algebra, University of Oxford, 1969. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[2]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.  Google Scholar

[3]

I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., 59 (1946), 54-106.  doi: 10.1090/S0002-9947-1946-0016094-3.  Google Scholar

[4]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.  Google Scholar

[5]

S. T. Dougherty, J. Kim and H. Kulosman, MDS codes over finite principal ideal rings, MDS Codes Over Finite Principal Ideal Rings, 50 (2009), 77-92.  doi: 10.1007/s10623-008-9215-5.  Google Scholar

[6]

S. T. Dougherty, S. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrete. Math., 305 (2005), 123-135.  doi: 10.1016/j.disc.2005.08.004.  Google Scholar

[7]

S. T. Dougherty, H. Liu and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ., 53 (2011), 39-53.   Google Scholar

[8]

S. T. Dougherty and H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia, 31B (2011), 331-343.  doi: 10.1016/S0252-9602(11)60233-6.  Google Scholar

[9]

S. T. Dougherty and Y. H. Park, Codes over the $p$-adic integers, Des. Codes Cryptogr., 39 (2006), 65-80.  doi: 10.1007/s10623-005-2542-x.  Google Scholar

[10]

D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[11]

E. E. Enochs and O. M. G. Jenda, Relative Homolodical Algebra, Walter de Gruyter, 2000. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[12]

K. Guenda and T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields Appl., 18 (2012), 1061-1075.  doi: 10.1016/j.ffa.2012.09.003.  Google Scholar

[13]

S. Jean-Pierre, Local Fields, Berlin, New York, 1980. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[14]

F. J. Makwilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[15]

H. Matsumura, Commutative Ring Theory, Cambridge, 1989. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[16]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[17]

K. R. McLean, Commutative Artinian principal ideal rings, Proc. London Math. Soc., 26 (1973), 249-272.  doi: 10.1112/plms/s3-26.2.249.  Google Scholar

[18]

K. Samei and S. Mahmoudi, Cyclic $R$-additive codes, Discrete Math., 340 (2017), 1657-1668.  doi: 10.1016/j.disc.2016.11.007.  Google Scholar

[19]

K. Samei and S. Mahmoudi, Singleton Bundes for $R$-additive codes, Adv. Math. Commun., 12 (2018), 107-114.  doi: 10.3934/amc.2018006.  Google Scholar

[20]

P. Solé, Open problems 2: Cyclic codes over rings and $p$-adic fields, in (G. Cohen and J. Wolfmann eds.) Coding Theory and Applications, Lect. Notes Comp. Sci., 338, Springer-Verlag, 1988. doi: 10.1007/BFb0019872.  Google Scholar

[1]

Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $ R_{k,m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020097
[2]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029
[3]

Hideaki Takagi. Times until service completion and abandonment in an M/M/$ m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028
[4]

Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123
[5]

Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016
[6]

Lin Yi, Xiangyong Zeng, Zhimin Sun. On finite length nonbinary sequences with large nonlinear complexity over the residue ring $ \mathbb{Z}_{m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020091
[7]

Zalman Balanov, Yakov Krasnov. On good deformations of $ A_m $-singularities. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122
[8]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079
[9]

Habibul Islam, Om Prakash, Patrick Solé. $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020094
[10]

Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051
[11]

Van Hoang Nguyen. A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5755-5764. doi: 10.3934/dcds.2020244
[12]

Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096
[13]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033
[14]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
[15]

Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020135
[16]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133
[17]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021
[18]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
[19]

Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030
[20]

Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $ n $-slice algebras and related algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021009

2019 Impact Factor: 0.734

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences