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doi: 10.3934/amc.2020122
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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 Early access 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

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